# 4: Groups III - Mathematics

• Tom Denton (Fields Institute/York University in Toronto)

For n > 1 , the group An is the commutator subgroup of the symmetric group Sn with index 2 and has therefore n!/2 elements. It is the kernel of the signature group homomorphism sgn : Sn → <1, −1>explained under symmetric group.

The group An is abelian if and only if n ≤ 3 and simple if and only if n = 3 or n ≥ 5 . A5 is the smallest non-abelian simple group, having order 60, and the smallest non-solvable group.

The group A4 has the Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions < (), (12)(34), (13)(24), (14)(23) >, that is the kernel of the surjection of A4 onto A3 = Z3 . We have the exact sequence V → A4 → A3 = Z3 . In Galois theory, this map, or rather the corresponding map S4 → S3 , corresponds to associating the Lagrange resolvent cubic to a quartic, which allows the quartic polynomial to be solved by radicals, as established by Lodovico Ferrari.

As in the symmetric group, any two elements of An that are conjugate by an element of An must have the same cycle shape. The converse is not necessarily true, however. If the cycle shape consists only of cycles of odd length with no two cycles the same length, where cycles of length one are included in the cycle type, then there are exactly two conjugacy classes for this cycle shape (Scott 1987, §11.1, p299).

• The two permutations (123) and (132) are not conjugates in A3, although they have the same cycle shape, and are therefore conjugate in S3.
• The permutation (123)(45678) is not conjugate to its inverse (132)(48765) in A8, although the two permutations have the same cycle shape, so they are conjugate in S8.

An is generated by 3-cycles, since 3-cycles can be obtained by combining pairs of transpositions. This generating set is often used to prove that An is simple for n ≥ 5 .

n Aut(An) Out(An)
n ≥ 4, n ≠ 6 Sn Z2
n = 1, 2 Z1 Z1
n = 3 Z2 Z2
n = 6 S6 ⋊ Z2 V = Z2 × Z2

For n > 3 , except for n = 6 , the automorphism group of An is the symmetric group Sn, with inner automorphism group An and outer automorphism group Z2 the outer automorphism comes from conjugation by an odd permutation.

For n = 1 and 2, the automorphism group is trivial. For n = 3 the automorphism group is Z2, with trivial inner automorphism group and outer automorphism group Z2.

The outer automorphism group of A6 is the Klein four-group V = Z2 × Z2 , and is related to the outer automorphism of S6. The extra outer automorphism in A6 swaps the 3-cycles (like (123)) with elements of shape 3 2 (like (123)(456) ).

There are some exceptional isomorphisms between some of the small alternating groups and small groups of Lie type, particularly projective special linear groups. These are:

• A4 is isomorphic to PSL2(3) [1] and the symmetry group of chiral tetrahedral symmetry.
• A5 is isomorphic to PSL2(4), PSL2(5), and the symmetry group of chiral icosahedral symmetry. (See [1] for an indirect isomorphism of PSL2(F5) → A5 using a classification of simple groups of order 60, and here for a direct proof).
• A6 is isomorphic to PSL2(9) and PSp4(2)'.
• A8 is isomorphic to PSL4(2).

More obviously, A3 is isomorphic to the cyclic group Z3, and A0, A1, and A2 are isomorphic to the trivial group (which is also SL1(q) = PSL1(q) for any q).

Cayley table of the alternating group A4
Elements: The even permutations (the identity, eight 3-cycles and three double-transpositions (double transpositions in boldface))

A5 is the group of isometries of a dodecahedron in 3-space, so there is a representation A5 → SO3(R) .

In this picture the vertices of the polyhedra represent the elements of the group, with the center of the sphere representing the identity element. Each vertex represents a rotation about the axis pointing from the center to that vertex, by an angle equal to the distance from the origin, in radians. Vertices in the same polyhedron are in the same conjugacy class. Since the conjugacy class equation for A5 is 1 + 12 + 12 + 15 + 20 = 60 , we obtain four distinct (nontrivial) polyhedra.

The vertices of each polyhedron are in bijective correspondence with the elements of its conjugacy class, with the exception of the conjugacy class of (2,2)-cycles, which is represented by an icosidodecahedron on the outer surface, with its antipodal vertices identified with each other. The reason for this redundancy is that the corresponding rotations are by π radians, and so can be represented by a vector of length π in either of two directions. Thus the class of (2,2)-cycles contains 15 elements, while the icosidodecahedron has 30 vertices.

The two conjugacy classes of twelve 5-cycles in A5 are represented by two icosahedra, of radii 2 π /5 and 4 π /5, respectively. The nontrivial outer automorphism in Out(A5) ≃ Z2 interchanges these two classes and the corresponding icosahedra.

It can be proved that the 15 puzzle, a famous example of the sliding puzzle, can be represented by the alternating group A15, [2] because the combinations of the 15 puzzle can be generated by 3-cycles. In fact, any 2k − 1 sliding puzzle with square tiles of equal size can be represented by A2k−1.

A4 is the smallest group demonstrating that the converse of Lagrange's theorem is not true in general: given a finite group G and a divisor d of | G |, there does not necessarily exist a subgroup of G with order d: the group G = A4 , of order 12, has no subgroup of order 6. A subgroup of three elements (generated by a cyclic rotation of three objects) with any distinct nontrivial element generates the whole group.

For all n > 4 , An has no nontrivial (that is, proper) normal subgroups. Thus, An is a simple group for all n > 4 . A5 is the smallest non-solvable group.

The group homology of the alternating groups exhibits stabilization, as in stable homotopy theory: for sufficiently large n, it is constant. However, there are some low-dimensional exceptional homology. Note that the homology of the symmetric group exhibits similar stabilization, but without the low-dimensional exceptions (additional homology elements).

### H1: Abelianization Edit

The first homology group coincides with abelianization, and (since An is perfect, except for the cited exceptions) is thus:

This is easily seen directly, as follows. An is generated by 3-cycles – so the only non-trivial abelianization maps are An → Z3, since order-3 elements must map to order-3 elements – and for n ≥ 5 all 3-cycles are conjugate, so they must map to the same element in the abelianization, since conjugation is trivial in abelian groups. Thus a 3-cycle like (123) must map to the same element as its inverse (321), but thus must map to the identity, as it must then have order dividing 2 and 3, so the abelianization is trivial.

For n < 3 , An is trivial, and thus has trivial abelianization. For A3 and A4 one can compute the abelianization directly, noting that the 3-cycles form two conjugacy classes (rather than all being conjugate) and there are non-trivial maps A3 ↠ Z3 (in fact an isomorphism) and A4 ↠ Z3 .

### H2: Schur multipliers Edit

The Schur multipliers of the alternating groups An (in the case where n is at least 5) are the cyclic groups of order 2, except in the case where n is either 6 or 7, in which case there is also a triple cover. In these cases, then, the Schur multiplier is (the cyclic group) of order 6. [3] These were first computed in (Schur 1911).

H2(An, Z) = Z1 for n = 1, 2, 3 H2(An, Z) = Z2 for n = 4, 5 H2(An, Z) = Z6 for n = 6, 7 H2(An, Z) = Z2 for n ≥ 8.

## 4: Groups III - Mathematics

The multiplication tables given below cover the groups of order 10 or less. That is, any group of order 2 through 10 is isomorphic to one of the groups given on this page. The reader needs to know these definitions: group, cyclic group, symmetric group, dihedral group, direct product of groups, subgroup, normal subgroup. The quaternion group is discussed in Example 3.3.7. There are more group tables at the end of Section 7.10.

Subgroups:
order 2: <1,a>
order 1:

#### C3, the cyclic group of order 3

Subgroups:
order 3: <1,a,a 2 >
order 1:

#### C4, the cyclic group of order 4

Elements:
order 4: a, a 3
order 2: a 2

Subgroups:
order 4: <1,a,a 2 ,a 3 >
order 2: <1,a 2 >
order 1:

#### C5, the cyclic group of order 5

Elements:
order 5: a, a 2 , a 3 , a 4

#### C6, the cyclic group of order 6

Elements:
order 6: a, a 5
order 3: a 2 , a 4
order 2: a 3

#### S3, the symmetric group on three elements

Elements:
order 3: a, a 2
order 2: b, ab, a 2 b

#### C7, the cyclic group of order 7

Elements:
order 7: a, a 2 , a 3 , a 4 , a 5 , a 6

#### C8, the cyclic group of order 8

Elements:
order 8: a, a 3 , a 5 , a 7
order 4: a 2 , a 6
order 2: a 4

#### C4 x C2, the direct product of a cyclic group of order 4 and a cyclic group of order 2

Elements:
order 4: a, a 3 , ab, a 3 b
order 2: a 2 , b, a 2 b
order 1: 1

#### C2 x C2 x C2, the direct product of 3 cyclic groups of order 2

Elements:
order 2: a, b, ab, c, ac, bc, abc

#### D4, the dihedral group of order eight

Elements:
order 4: a, a 3
order 2: a 2 , b, ab, a 2 b, a 3 b

#### Q, the quaternion group (of order eight)

Elements:
order 4: a, a 3 , b, ab, a 2 b, a 3 b
order 2: a 2

Here are several different patterns for the multiplication table of the quaternion group, using the cross product of unit vectors i, j, k:

Elements:
order 4: i, -i, j, -j, k, -k
order 2: -1

#### C9, the cyclic group of order 9

Elements:
order 9: a, a 2 , a 4 , a 5 , a 6 , a 7
order 3: a 3 , a 6

#### C3 x C3, the direct product of two cyclic groups of order 3

Elements:
order 3: a, a 2 , b, ab, a 2 b, b 2 , ab 2 , a 2 b 2

#### C10, the cyclic group of order 10

Elements:
order 10: a, a 3 , a 7 , a 9
order 5: a 2 , a 4 , a 6 , a 8
order 2: a 5

#### D5, the dihedral group of order ten

Elements:
order 5: a, a 2 , a 3 , a 4
order 2: b, ab, a 2 b, a 3 b, a 4 b

## 4: Groups III - Mathematics

(ii) Associativity: For all a,b,c G, we have

(iii) Identity: There exists an identity element e G such that

(iv) Inverses: For each a G there exists an inverse element a -1 G such that

We will usually simply write ab for the product a · b.

3.1.6. Proposition. (Cancellation Property for Groups) Let G be a group, and let a,b,c G. (a) If ab=ac, then b=c.

(b) If ac=bc, then a=b. 3.1.8. Definition. A group G is said to be abelian if ab=ba for all a,b G.

3.1.9. Definition. A group G is said to be a finite group if the set G has a finite number of elements. In this case, the number of elements is called the order of G, denoted by |G|.

3.2.7. Definition. Let a be an element of the group G. If there exists a positive integer n such that a n = e, then a is said to have finite order , and the smallest such positive integer is called the order of a, denoted by o(a).
If there does not exist a positive integer n such that a n = e, then a is said to have infinite order .

3.2.1. Definition. Let G be a group, and let H be a subset of G. Then H is called a subgroup of G if H is itself a group, under the operation induced by G.

3.2.2. Proposition. Let G be a group with identity element e, and let H be a subset of G. Then H is a subgroup of G if and only if the following conditions hold: (i) ab H for all a,b H

(iii) a -1 H for all a H. 3.2.10. Theorem. (Lagrange) If H is a subgroup of the finite group G, then the order of H is a divisor of the order of G.

3.2.11. Corollary. Let G be a finite group of order n. (a) For any a G, o(a) is a divisor of n.

(b) For any a G, a n = e. Example 3.2.12. (Euler's theorem) Let G be the multiplicative group of congruence classes modulo n. The order of G is given by (n), and so by Corollary 3.2.11, raising any congruence class to the power (n) must give the identity element.

3.2.12. Corollary. Any group of prime order is cyclic.

3.4.1. Definition. Let G1 and G2 be groups, and let : G1 -> G2 be a function. Then is said to be a group isomorphism if (i) is one-to-one and onto and

(ii) (ab) = (a) (b) for all a,b G1. In this case, G1 is said to be isomorphic to G2, and this is denoted by G1 G2.

3.4.3. Proposition. Let : G1 -> G2 be an isomorphism of groups. (a) If a has order n in G1, then (a) has order n in G2.

(b) If G1 is abelian, then so is G2.

(c) If G1 is cyclic, then so is G2.

## Cyclic groups

is called the cyclic subgroup generated by a.
The group G is called a cyclic group if there exists an element a G such that G=<a>. In this case a is called a generator of G.

3.2.6 Proposition. Let G be a group, and let a G. (a) The set <a> is a subgroup of G.

(b) If K is any subgroup of G such that a K, then <a> K. 3.2.8. Proposition. Let a be an element of the group G. (a) If a has infinite order, and a k = a m for integers k,m, them k=m.

(b) If a has finite order and k is any integer, then a k = e if and only if o(a) | k.

(c) If a has finite order o(a)=n, then for all integers k, m, we have

a k = a m if and only if k m (mod n).

Furthermore, |<a>|=o(a). Corollaries to Lagrange's Theorem (restated): (a) For any a G, o(a) is a divisor of |G|.

(b) For any a G, a n = e, for n = |G|.

(c) Any group of prime order is cyclic. 3.5.1. Theorem. Every subgroup of a cyclic group is cyclic.

3.5.2 Theorem. Let G cyclic group. (a) If G is infinite, then G Z .

(b) If |G| = n, then G Z n. 3.5.3. Proposition. Let G = <a> be a cyclic group with |G| = n. (a) If m Z , then <a m > = <a d >, where d=gcd(m,n), and a m has order n/d.

(b) The element a k generates G if and only if gcd(k,n)=1.

(c) The subgroups of G are in one-to-one correspondence with the positive divisors of n.

(d) If m and k are divisors of n, then <a m > <a k > if and only if k | m. 3.5.6. Definition. Let G be a group. If there exists a positive integer N such that a N =e for all a G, then the smallest such positive integer is called the exponent of G.

3.5.7. Lemma. Let G be a group, and let a,b G be elements such that ab = ba. If the orders of a and b are relatively prime, then o(ab) = o(a)o(b).

3.5.8. Proposition. Let G be a finite abelian group. (a) The exponent of G is equal to the order of any element of G of maximal order.

(b) The group G is cyclic if and only if its exponent is equal to its order.

## Permutation groups

3.1.5. Proposition. If S is any nonempty set, then Sym(S) is a group under the operation of composition of functions.

2.3.5. Theorem. Every permutation in Sn can be written as a product of disjoint cycles. The cycles that appear in the product are unique.

2.3.8 Proposition. If a permutation in Sn is written as a product of disjoint cycles, then its order is the least common multiple of the lengths of its cycles.

3.6.1. Definition. Any subgroup of the symmetric group Sym(S) on a set S is called a permutation group or group of permutations .

3.6.2. Theorem. (Cayley) Every group is isomorphic to a permutation group.

3.6.3. Definition. Let n > 2 be an integer. The group of rigid motions of a regular n-gon is called the nth dihedral group , denoted by Dn.

We can describe the nth dihedral group as

subject to the relations o(a) = n, o(b) = 2, and ba = a -1 b.

2.3.11. Theorem. If a permutation is written as a product of transpositions in two ways, then the number of transpositions is either even in both cases or odd in both cases.

2.3.12. Definition. A permutation is called even if it can be written as a product of an even number of transpositions, and odd if it can be written as a product of an odd number of transpositions.

3.6.4. Proposition. The set of all even permutations of Sn is a subgroup of Sn.

3.6.5. Definition. The set of all even permutations of Sn is called the alternating group on n elements, and will be denoted by An.

## Other examples

3.1.10. Definition. The set of all invertible n × n matrices with entries in R is called the general linear group of degree n over the real numbers, and is denoted by GLn( R ).

3.1.11. Proposition. The set GLn( R ) forms a group under matrix multiplication.

3.3.3. Definition. Let G1 and G2 be groups. The set of all ordered pairs (x1,x2) such that x1 G1 and x2 G2 is called the direct product of G1 and G2, denoted by G1 × G2.

3.3.4. Proposition. Let G1 and G2 be groups. (a) The direct product G1 × G2 is a group under the multiplication defined for all
(a1,a2), (b1,b2) G1 × G2 by

(b) If the elements a1 G1 and a2 G2 have orders n and m, respectively, then in
G1 × G2 the element (a1,a2) has order lcm[n,m]. 3.3.5. Definition. Let F be a set with two binary operations + and · with respective identity elements 0 and 1, where 1 is distinct from 0. Then F is called a field if (i) the set of all elements of F is an abelian group under +

(ii) the set of all nonzero elements of F is an abelian group under ·

(iii) a · (b+c) = a · b + a · c for all a,b,c in F. 3.3.6. Definition. Let F be a field. The set of all invertible n × n matrices with entries in F is called the general linear group of degree n over F, and is denoted by GLn(F).

3.3.7. Proposition. Let F be a field. Then GLn(F) is a group under matrix multiplication.

3.4.5. Proposition. If m,n are positive integers such that gcd(m,n)=1, then

Example. 3.3.7. (Quaternion group)
Consider the following set of invertible 2 × 2 matrices with entries in the field of complex numbers.

then we have the identities

j i = - k , k j = - i , i k = - j .

These elements form a nonabelian group Q of order 8 called the quaternion group, or group of quaternion units.

## 4: Groups III - Mathematics

Now let us explore some infinite groups.

1. The set of integers (resp. rational numbers, real numbers) forms a group under +, denoted by Z (resp. Q + , R + ).
2. The set of non-zero rational (resp. real, complex) numbers forms a group under ×, denoted by Q * (resp. R * , C * ).
3. The set of positive rational (resp. real) numbers is a group under ×, denoted by Q >0 (resp. R >0 ).
4. The set of rational numbers is a group under the operation a * b = a + b – 5. Exercise: what is the identity, and the inverse of a?
5. The set of invertible n× n matrices with real entries is a group under multiplication. This is denoted by GL(n, R) – the general linear group of degree n.
6. The set of n× n matrices with real entries and determinant = 1, forms a group under multiplication. This is denoted by SL(n, R) – the special linear group of degree n.
7. Let X be any set. The set of bijections f : XX forms a group SXunder composition: this generalises the case of Snwhich takes the case of X = <1, 2, …, n>. Note that if X is infinite, then so is SX.
8. The set of all invertible affine transformations of the real line R is a group. Here, an affine transformation is a map f : RR given by f(x) = ax + b for some fixed real a, b (and a ≠ 0). Notice that composition and inverse of affine transformations are still affine.

Examples 1-4 are abelian groups while the remaining are all non-abelian.

Exercise: prove that Z is not isomorphic to Q + .

Exercise: prove that R is isomorphic to R >0 .

Exercise (tricky): prove that Q + is not isomorphic to R + .

Finally, we have the following exercises.

Exercise (easy) : prove that if G is abelian, and g, g’ are elements of G with finite order, then gg’ has finite order.

Exercise (tricky): find an infinite group such that every element has order at most 2.

Exercise (hard): find a group G with elements g, g’ such that g and g’ have order 2, but gg’ has infinite order.

## Mathematics - Class III Resources

The following list of online resources are provided as a sample of curated resources. New resources shall be added as we come across interesting and relevant online materials.

#### 1. Let's Learn Fractions

Description: This video contains explanation on fraction (numerator and denominator).

Core Concepts: III-A2 simple fractions.

#### 2. Decimal Models

Description: This video contains explanation on concept of decimal tenths.

Core Concepts: III-A3 decimal tenths.

Description: This video shows how to add 3-digit numbers with regrouping.

Core Concepts: III-A5 add 3-digit whole numbers and III-A7 add and subtract 3-digit numbers mentally.

#### 4. Subtraction with Regrouping 3 Digit

Description: This video shows how to subtract 3-digit numbers with by regrouping.

Core Concepts: III-A6 subtract 3-digit whole numbers and III-A7 add and subtract 3-digit numbers mentally.

#### 5. Learning to Multiply using Multiplication Strategies

Description: This video contains explanation on four multiplication strategies (array, equal groups, repeated addition and number lines).

Core Concepts: III-A8 multiplication.

#### 6. Multiplying: 2 Digits Times 1 Digit

Description: This video shows how to multiply a 2-digit number by a 1-digit number.

Core Concepts: III-A8 multiplication.

#### 7. Division Using Equal Groups

Description: This video shows how to divide using equal grouping.

Core Concepts: III-A11 division meanings.

#### 8. Division as Equal Sharing

Description: This video explains division as equal sharing.

Core Concepts: III-A11 division meanings.

#### 9.Division as Repeated Subtraction

Description: This video explains division as repeated subtraction.

Core Concepts: III-A11 division meanings.

#### 10. Multiplication and Division Fact Families

Description: This video explains the link between multiplication and division fact families.

Core Concepts: III-A12 multiplication and division.

#### 11. Angles - Types and Definition (Mathematics for Kids)

Description: This video contains explanation of definition of angles.

Core Concepts: III-C1 angles.

#### 12. Understanding MM, CM, M, and KM

Description: This video explains four units of measuring length (mm, cm, m and km).

Core Concepts: III-C2 Length: Relationship among different units of measuring length.

#### 13. Mathematics Key Stage 1 : Area and Perimeter

Description: This video contains demonstration of measuring perimeter in cm.

Core Concepts: III-C2 Length: Relationship among different units of measuring length.

#### 14. Litres and Millilitres | Maths for Kids | Grade 3 | Periwinkle

Description: This video explains the units for measuring capacity (L and ml).

Core Concepts: III-C3 capacity: Measuring capacity in litre and measuring capacity in millilitre.

#### 15. Measuring Mass in Grams

Description: This video shows measuring mass of smaller objects in grams.

Core Concepts: III-C4 mass: Measuring mass in kilogram and measuring mass in gram.

#### 16. Grams and Kilograms | Maths for Kids | Grade 3 | Periwinkle

Description: This video explains conversion of kilogram to gram.

Core Concepts: III-C4 mass: Measuring mass in kilogram and measuring mass in gram.

#### 17. Mathematics Key Stage 1 : Area and Perimeter

Description: This video contains demonstration of measuring area using square centimeter.

Core Concepts: III-C5 area.

#### 18. Mathematics Key Stage 1 : Reading Time

Description: This video contains demonstration of reading time on analog and digitals clocks and converting hours to minutes.

Core Concepts: III-C6 measuring time and reading analog and digital clocks relation among different units of time.

#### 19. Math Antics - Polygons

Description: This video explains features of polygons.

Core Concepts: III-D1 polygons, III-D2 squares & rectangles, and III-D3 parallelograms.

#### 20. Regular & Irregular Polygons

Description: This video shows the differences between Regular and Irregular Polygons.

Core Concepts: III-D1 polygons, III-D2 squares & rectangles, and III-D3 parallelograms.

#### 21. Quadrilaterals (by Math Antics)

Description: This video explains properties of different quadrilaterals.

Core Concepts: III-D1 polygons, III-D2 squares & rectangles, and III-D3 parallelograms.

#### 22. 3D Figures - Prisms and Pyramids | Math | Grade-3,4 | TutWay |

Description: This video shows the attributes of prisms and pyramids.

Core Concepts: III-D4 prisms & pyramids.

#### 23. Combining and Subdividing Shapes - Math, Grade 5, Unit 8, Video 9

Description: This video contains demonstration on how to combine polygons.

Core Concepts: II-D5 combining two or more shapes and III-D7 similar and congruent shapes.

#### 24. Difference Between Similar and Congruent Figures

Description: This video explains the difference between similar or congruent shapes.

Core Concepts: II-D5 combining two or more shapes and III-D7 similar and congruent shapes.

#### 25. Shapes: Flips, Slides and Turns

Description: This video shows how to flip, slide and turn a shape.

Core Concepts: III-D6 turns, slides and flip of 2-D shapes.

#### 26. Collecting and Organizing Data

Description: This video shows how to collect data and organize data using tally chart.

Core Concepts: III-E1 data collection.

#### 27. Pictograph | Maths for Kids | Grade 4 | Periwinkle

Description: This video shows how to create and interpret a pictograph.

Core Concepts: III-E2 pictograph.

#### 28. Graphs - Bar Graphs | Math | Grade-4,5 | Tutway |

Description: This video demonstrates how to create bar graphs both vertically and horizontally and also to interpret bar graphs.

## Partially Ordered Set (POSET)

A partially ordered set consists of a set with a binary relation which is reflexive, antisymmetric and transitive. "Partially ordered set" is abbreviated as POSET.

### Examples

The set of real numbers under binary operation less than or equal to $(le)$ is a poset.

Let the set $S = lbrace 1, 2, 3 brace$ and the operation is $le$

The relations will be $lbrace(1, 1), (2, 2), (3, 3), (1, 2), (1, 3), (2, 3) brace$

This relation R is reflexive as $lbrace (1, 1), (2, 2), (3, 3) brace in R$

This relation R is anti-symmetric, as

$lbrace (1, 2), (1, 3), (2, 3) brace in R and lbrace (1, 2), (1, 3), (2, 3) brace ¬in R$

This relation R is also transitive as $lbrace (1,2), (2,3), (1,3) brace in R$.

The vertex set of a directed acyclic graph under the operation ‘reachability’ is a poset.

## MULTIPLICATION

Description: Help your county and state rise to the top of the leaderboards in the Great American Multiplication Challenge. PLEASE NOTE: On or around December 22, we will be resetting the statistics for this game and changing the format to match those of the Great American Addition and Subtraction Challenges.

Format: Printable Activity

The Legend of Multiplico - A Multiplication and Division Adventure Game

Description: The evil Horrefedous has four mythical creatures in his grips yet again! This time he has hidden or imprisoned them in a network of underground rooms, full of enemies. You must defeat these enemies with your multiplication and division wizardry, earning precious neurons as you go. After all of your adventuring and wizardry, you must face Horrefedous in an all-or-none multiplication attack in order to save the creatures. Be fast with your multiplication, but most importantly, be accurate. Wrong answers will drain your life and cost you neurons.

Multiplication Pal - Online Multiplication Simulation

Description: This amazing tools allows students to complete small or large multiplication, step-by-step, in an interview format. Students can even enter their own problem! This is a MUST try.

CC Standards: 4.NBT.B.5, 4.NBT.B.6

Description: Drag 'N' Drop Math is an online workshop in which students can easily complete multi-digit addition, subtraction (with regrouping), multiplication, and division problems, using big and small draggable numbers. The workshop is totally customizable and gives immediate feedback. This is one of the top ten most popular programs on mrnussbaum.com

CC Standards: 2.NBT.B.5, 2.NBT.B.6, 2.NBT.B.7, 2.NBT.B.8, 3.OA.A.4, 3.OA.C.7, 3.NBT.A.2, 3.NBT.A.3

Fun Multiplication Games - From ComputerMice

Description: Need to practice multiplication facts? Fun Multiplication Games from Computer Mice is the perfect solution. You can practice multiplication fluency by playing any of 15 embedded games including target practice games, ninja baby games, spinning wheel games, and many more. Look throughout our games, math, and language arts section for more games from Computer Mice soon.

The Multiplication Zombies of the Brittany Graveyard - Online Game

Description: The Zombies of the Brittany Graveyard have been a scourge upon the village for many years - terrifying those who wish to visit the graves of their loved ones (animations). Recently, the townspeople came together to call upon you, the world's foremost zombie exterminator to bring light once again to their cemetery by defeating the zombies. Use your unique and high-powered multiplication skills to fling your devastating jack-o-lanterns at the hapless zombies. If you can clear each of the five points of the cemetery of the horrid zombies, you'll succeed in your task of liberating the cemetery and will receive a key to the village of Brittany.

Factor Family Reunion - Online Game

Description: The Factors are having a family reunion and YOU are hosting it. It’s your job to make sure each member of the factor family is seated at the correct table, or, you’ll hear it from them! Simply drag and drop each factor to its correct table. If you can get them all, you can print out a portrait of the entire at their reunion.

CC Standards: 3.OA.C.7, 4.OA.A.1, 4.OA.B.4

Around the World - Online Multiplication Game

Description: Around the World is a fun multiplication game based on the timeless classic classroom game where students go "Around the World" if they can defeat their classmates in a game of multiplication flash cards. In the online versions, students face fictional students from other countries, thereby integrating the game well with geography. Students win if they can defeat all 20 students. The problem is, “students” from different countries answer the flash cards at different speeds. Some may take ten seconds, while others may take only 4 or 5 seconds. In this way, the game simulates the real game where some kids are quicker with their facts than others.

Description: First choose your skill to practice (addition, subtraction, multiplication, or division). Then, choose the numbers you want to practice. Finally, indicate whether or not to allow negative numbers. For example, if you want to practice adding 1, 2, and 3, click on the 1 bubble, the 2 bubble, and the 3 bubble. Finally, set the countdown to however many seconds you want and see how many problems you can correctly answer, or, set an attainment goal, and see how long it takes you to reach your goal! If you reach your goal, you can print out your very own certificate of achievement.

CC Standards: K.OA.A.5, 1.OA.C.6, 2.OA.B.2, 3.OA.C.7, 6.NS.C.6.A, 7.NS.A.1

Factorization Forest - Online Game

Description: Factorization Forest is a game in which students can practice their prime factorization skills. Students can choose to build a forest without a timer using their prime factorization skills, or, can play a game in which they try to populate a river valley with as many trees as possible in three minutes using their prime factorization skills.To play, choose the game type and select the type of tree you would like to grow. Then, turn your attention to the number that appears at the top of the screen. This will be the number you will “factorize”. Click the ” + ” button to begin building your factor tree. Continue clicking the ” + ” buttons that appear until your are left with only prime numbers. Then, count of the prime numbers to form your factorization. To type in your factorization, find the space toward the bottom of the screen in which you can enter a number and use the ” + ” to enter other numbers. Use the dotted lines positioned to the upper right of each number to specify exponents. When you are satisfied with your factorization, click the ” ? ” button. If you are correct, you will see your tree grow. If you are playing the create a forest version of the game, you can move your tree to any place on the picture. If you are playing the timed version, the tree will remain in a fixed position.

Crossing Math Canyon - Online Game

Description: For hundreds of years, the famous but elusive Golden Medallion of Math Canyon has proved unreachable and deadly for dozens of brave explorers who have tried crossing the invisible bridge for the purposes of obtaining it. Now, it is your turn. The Bridge that crosses Math Canyon will form plank by plank as you step on the correct planks. Red Hawk, an ancient Pueblo warrior, will guide you along the way. First, step on planks that are multiples of two (use your counting by two skills). Then, all multiples of three, four, five, and so on until you complete multiples of nine. Then, and only then, can you uncover the long lost golden medallion of Math Canyon. Be careful, however, stepping on the wrong plank will send you plummeting in the river below.

The Ultimate Teacher's Lounge - Online Game

Description: Why wait until Teacher Appreciation Week to honor your teacher? Use your amazing flash card skills to earn as many “neurons” as possible. Use the “tab” key to move from flash card to flash card. Then, spend your “neurons” at the Teacher’s Lounge Store and score a hot tub, dance floor, big screen, popcorn machine and much more to make your teacher’s lounge the best in history.

CC Standards: 1.OA.C.6, 2.OA.B.2, 3.OA.C.7, K.OA.A.5

Becoming Lord Voldemath - Online Game

Description: This game allows students customized practice with specific "tables" in addition, subtraction, multiplication, and division. Students battle "wizards" to answer problems quickest in each of five 90-second rounds. If the students has a higher score than the wizard, he or she moves on to the next round and gains a new "power." Students LOVE this game which serves as great quick math reinforcement.

CC Standards: K.OA.A.5, 1.OA.C.6, 2.OA.B.2, 3.OA.C.7

Description: This super-fast paced game requires students to ski through the gates that complete an equation, but to avoid those that make the equation incorrect. For example, if a student chooses x 8 to practice, he or she would ski through gates that show 2 and 16, but around gates that show 4 and 30. The game is customizable and allows players to choose the operation and the specific numbers.

Description: Math Machine is a VISUAL tool for teaching addition, subtraction, multiplication, fractions, division, or place value. Students are empowered by spinning wheels that determine numbers in the problems! See instructional video for more information.

CC Standards: 1.OA.A.1, 1.OA.A.2, 1.OA.B.3, 1.OA.C.5, 1.OA.C.6, 2.OA.A.1, 2.OA.B.2, 2.OA.C.3, 2.OA.C.4, 2.NBT.A.1, 2.NBT.B.5, 3.OA.A.1, 3.OA.A.2, 3.OA.C.7, 3.NF.A.3

Description: This is a fun football-themed math game where students rumble down the field using their addition, subtraction, and multiplication skills. Students play offense and defense!

Bowling Pin Math - Online Game

Description: Bowling Pin Math is an awesome game where students must determine which math problems (located on the bowling pins) have answers that are greater than or less than the target number. In this way, students must evaluate ten math problems at once, rather than just the standard way of evaluating one math problem at a time. Students must bowl ten frames and score as close to 100 as possible.

World Cup Math - Online Game

Description: This online soccer shootout requires students to choose a team and battle others in a round-of-16 using his or her addition, subtraction, multiplication, or division skills.

Description: You are in a math museum filled with some of the greatest matherpieces of all time, painted by the likes of Pablo Multiplicasso, Factorangelo, and many others. But No! A villain, the Confounder, has broken in and switched all of the titles to amuse himself. Someone needs to help! Can you? Look at all the matherpieces and figure out what the title of each is. Luckily, the artists always chose simple titles that reflected the meaning of each painting.

Tae Kwon Donuts - Online Game

Description: For many years the Tae Kwon Donuts and the Subninjas have fought against each other. Now, the subninjas have resorted to kidnapping the Tae Kwon Donuts’ Munchquins! You must use your addition, subtraction, multiplication, and division skills for both positive and negative numbers to identify the weak link among rows of horrifying subninjas to save the future generation of Tae Kwon Donuts! You must attack the subninja with the math problem that yields a different answer than the rest! Use the keyboard arrows to move your Tae Kwon Donut and the space bar to attack. As you progress through rooms of the castle, you will earn your colored belts and new attack modes! You can also earn a password to return to any room in the castle.

Description: Golden Path is appropriate for kids ages 7 – 10. The game requires students to choose an operation and play the role of a frog that must hop to the other side of the pond using lily pads labeled with math problems. Students must evaluate the math problems on two, three, four, or even five connected lily pads and must direct the frog to hop on the lily pad with the math problem that yields the greatest answer.

Description: This innovative game requires students to save seven members of a Royal Family from prison by using their order of operation skills to build stairways leading to their secret cells. Choose your character first and then begin solving the order of operations equation by clicking on the first number, then its operator, followed by the second number. For example, in a problem such 5 + 3 x 2 (6 – 4). The user should click on the 6 first, the “-” second, and the 4 last. Once two numbers and the operator have been clicked on, the program will isolate the problem to solve, in this case 6-4. If the student correctly solves the problem, a second step will appear with the shortened problem: 5 + 3 x 2 – 2. The user would then click on the 3 followed by the “x” and then the two. The resulting problem on the next step would be 5 + 6 – 2 and students would solve the last two problems before successfully saving the first of the royal family. This game is OUTSTANDING practice in order of operations and one of the only GAMES on the internet reinforcing this skills.

Description: Lunch Line is a fun (and funny) game in which students practice their fractions, decimals, and percentages ordering skills. Students must arrange the celebrities and historical figures in a lunch line based on the values floating on top of their heads from least to greatest. If students arrange all ten correctly, the lunch line will proceed smoothly to the cafeteria in a straight line and they’ll be able to print out a certificate showing the line leader. If figures are positioned incorrectly, the lunch line will stagger crookedly and inefficiently to the cafeteria, thereby angering the teacher.

Type: Math Game - Decimals Focus

Description: In Tipster, students player the role of restaurant manager who must calculate the tip amounts for his or her servers. This fun game involves calculating percentages of numbers and quality of service. Quality of service indicated by the customers determined percentage of total bill that constitutes tip. For example, the total bill at a table is $100.00, and the service was level was a "3," the customer pays 15% making the total bill$115. Very fun!

Zip-Lining Lunch Ladies - Multiplication by 2

Description: This super fun and create way to practice multiplication requires students to create zip lines for our adventurous lunch ladies by matching the product with its equation. Loads of fun. How fast can you get all eight lunch ladies to their places?

Zip-Lining Lunch Ladies - Multiplication by 3

Description: This super fun and create way to practice multiplication requires students to create zip lines for our adventurous lunch ladies by matching the product with its equation. Loads of fun. How fast can you get all eight lunch ladies to their places?

Zip-Lining Lunch Ladies - Multiplication by 4

Description: This super fun and create way to practice multiplication requires students to create zip lines for our adventurous lunch ladies by matching the product with its equation. Loads of fun. How fast can you get all eight lunch ladies to their places?

Zip-Lining Lunch Ladies - Multiplication by 6

Description: This super fun and create way to practice multiplication requires students to create zip lines for our adventurous lunch ladies by matching the product with its equation. Loads of fun. How fast can you get all eight lunch ladies to their places?

Zip-Lining Lunch Ladies - Multiplication by 7

Description: This super fun and create way to practice multiplication requires students to create zip lines for our adventurous lunch ladies by matching the product with its equation. Loads of fun. How fast can you get all eight lunch ladies to their places?

Zip-Lining Lunch Ladies - Multiplication by 5

Description: This super fun and create way to practice multiplication requires students to create zip lines for our adventurous lunch ladies by matching the product with its equation. Loads of fun. How fast can you get all eight lunch ladies to their places?

Zip-Lining Lunch Ladies - Multiplication by 8

Description: This super fun and create way to practice multiplication requires students to create zip lines for our adventurous lunch ladies by matching the product with its equation. Loads of fun. How fast can you get all eight lunch ladies to their places?

Zip-Lining Lunch Ladies - Multiplication by 9

Description: This super fun and create way to practice multiplication requires students to create zip lines for our adventurous lunch ladies by matching the product with its equation. Loads of fun. How fast can you get all eight lunch ladies to their places?

Zip-Lining Lunch Ladies - Multiplication by 11

Description: This super fun and create way to practice multiplication requires students to create zip lines for our adventurous lunch ladies by matching the product with its equation. Loads of fun. How fast can you get all eight lunch ladies to their places?

Zip-Lining Lunch Ladies - Multiplication by 12

Description: This super fun and create way to practice multiplication requires students to create zip lines for our adventurous lunch ladies by matching the product with its equation. Loads of fun. How fast can you get all eight lunch ladies to their places?

Conceptualizing Multiplication and Division Instructional Video

Description: This video explains uses examples to teach the concepts of division and multiplication.

Relating Addition to Multiplication - Online

Description: This activity requires students to envision multiplication as large addition problems. It gives immediate feedback.

Properties of Multiplication - Online

Description: This activity requires students to identify the multiplication property (associative, commutative, etc.) based on the equation. It gives immediate feedback.

Applying Properties of Multiplication - Online

Description: This activity requires students to apply the appropriate multiplication property (associative, commutative, etc.) to solve the equation. It gives immediate feedback.

Multiplication Statements - Online

Description: This activity requires students to think about multiplication in terms of division. For example, 56 is eight times as many as .

Description: This activity requires students to complete the in/out chart using multiplication.

Rectangular Arrays - Online

Description: This math drill requires students to answer questions about the rectangular arrays. Immediate feedback is given.

Making Rectangular Arrays

Description: This activity requires students to create arrays based on the questions. For example, how many is six rows of three?

Format: Printable Activity

Flash Card Multiplication - Around the World

Description: This activity will help students get ready to play Around the World. Students must write the correct products on the multiplication flash cards and color them the specified color.

Format: Printable Activity

Multiplying numbers by 8 - Online

Description: This is a simple online drill that requires students to multiply numbers by eight. It gives immediate feedback.

Multiplying Numbers by Eight

Description: This activity requires students to complete multiplication by eight flash cards.

Format: Printable Activity

Description: This is printable multiplication by 12 activity.

Format: Printable Activity

Multiplication by 12 - Online

Description: This is an interactive multiplication drill with flash cards. All requires students to multiply a number by 12. Immediate feedback is given.

Multiplication and the Number 24

Description: This activity features four two-digit by two-digit multiplication problems that each contain the number 24.

Format: Printable Activity

Description: This activity requires students to populate a venn diagram with factors of 24 and 36.

Format: Printable Activity

Description: This activity requires students to identify multiples of single-digit numbers.

Description: This activity requires students to identify factors of numbers.

Least Common Multiple - Online

Description: This activity will help students practice finding the least common multiple of two numbers. It gives immediate feedback.

Greatest Common Factor - Online

Description: This activity will help students practice finding the greatest common factor of two numbers. It gives immediate feedback.

Least Common Multiples and Greatest Common Factor Instructional Video

Description: This video explains how to find the least common multiple or greatest common factor of a pair of numbers.

Finding Missing Factors - Online

Description: This activity requires students to find the missing factors within multiplication problems. It gives immediate feedback.

Factoring Numbers Instructional Video

Description: This video explains how to find all of the factors of a number.

Description: This activity requires students to solve simple exponent problems.

Using Inequalities to Analyze Multi-digit Multiplication Equations

Description: This activity requires students to use signs of inequality to compare large multiplication problems. It gives immediate feedback.

Crossing Math Canyon Online Practice - Multiples of Numbers

Description: This fun online activity will help students learn to play Crossing Math Canyon. In the exercise, students must follow the path of the traveler and identify the "wrong step" taken.

Crossing Math Canyon - Multiples of Seven

Description: This activity will help students learn to play Crossing Math Canyon. Students must successfully cross the bridge by identifying all of the numbers that area multiples of seven.

Format: Printable Activity

Crossing Math Canyon - Multiples of Eight

Description: This activity will help students learn to play Crossing Math Canyon. Students must successfully cross the bridge by identifying all of the numbers that area multiples of eight.

Format: Printable Activity

Crossing Math Canyon - Multiples of Nine

Description: This activity will help students learn to play Crossing Math Canyon. Students must successfully cross the bridge by identifying all of the numbers that area multiples of nine.

Format: Printable Activity

What is the Missing Exponent? - Online

Description: This activity requires students to solve problems for the missing exponents.

Description: This fun activity combines math with world geography. It requires student yo use their multiplication skills to identify world nations. For example, a problem might say 5 x 8 = __________ (India -color red). Students would look on the map and find the "40" within the nation of India and color it red. Great integrated practice!

Format: Printable Activity

Speed Math Multiplication Practice - Online

Description: This online exercise reinforces multiplication facts and will also help students become accustomed to playing Speed Math.

Order Ops Demonstration Video

Description: This video will show you how to play Order Ops.

Golden Path Practice - Multiplication

Description: This activity will teach you how to use Golden Path. Which lily pad contains the multiplication problem with the largest product?

Word problems with addition, subtraction, multiplication, and division - Online

Description: This activity requires students to solve word problems with all four operations. It gives immediate feedback.

Multi-step Word problems with addition, subtraction, multiplication, and division - Online

Description: This activity requires students to solve multi-step word problems with all four operations. It gives immediate feedback.

Multiplying Two-digit Numbers by Ten - Online

Description: This activity requires students to complete the in/out chart using multiplication.

Multiplying Numbers Ending in Zero - Online

Description: This activity requires students to multiply numbers ending in zero. It gives immediate feedback.

Multiplying Decimals by Powers of Ten - Online

Description: This activity requires students to multiply decimals by powers of ten. It gives immediate feedback.

Estimating Products - Online

Description: This activity requires students to estimate the answers to multi-digit multiplication problems.

Estimating Products of Difficult Multiplication Problems - Online

Description: This activity requires students to estimate products of multiplication problems such as 67 x 54. It gives immediate feedback.

Word Problems Involving Estimating Products - Online

Description: This activity requires students to solve word problems in which they estimate the product of multi-digit numbers. It is multiple choice and immediate feedback is given.

Scientific Notation - Online

Description: This activity requires students to convert numbers in standard notation to their standard form.

Multiplying Two-digit by Two-digit Word Problems - Online

Description: This activity requires students to solve word problems with double-digit multiplication. It gives immediate feedback.

Intermediate Multiplication and Division Word Problems - Online

Description: This activity requires students to solve word problems that have multi-digit division and multiplication. Immediate feedback is provided.

Comparing Multiplication Equations Using Inequalities - Online

Description: This activity requires students to compare multiplication equations using signs of inequalities.

Fraction Workshop - Online

Description: Fraction Workshop is an amazing drag and drop application that allows students to complete any kind of fraction operation in an online stage with tools to help them. Fraction workshop allows users to practice ordering, reducing, adding, subtracting, multiplying, and dividing fractions and mixed numbers. Our drag and drop system makes ordering and organizing numbers easy. Choose the number of problems to practice, the specific skill to practice and click “begin”. Work the problem on the stage and drag and drop the correct numbers to the answer box. The system will indicate immediately whether or not your answer is correct. Printout a score summary when you are finished. Students can use the calculator tool or the visualize tool to help them work on the problems. The visualize tool turns the particular math problem into a picture. This helps students to better “see” the problem.

CC Standards: 3.NF.A.3, 4.NF.A.1, 4.NF.A.2, 4.NF.A.3, 4.NF.B.4, 4.NF.C.5, 5.NF.A.1, 5.NF.B.3, 5.NF.B.4, 5.NF.B.7

Fractions - Fractions of Numbers - Online

Description: This simple activity requires students to calculate fractions of numbers. For example, "What is 1/4 of 16?"

Fractions - Finding a Common Denominator - Online

Description: This activity requires students to practice finding the common denominator. It gives immediate feedback.

Multiplying Fractions by Whole Numbers - Online

Description: The activity helps students practice multiplying whole numbers by fractions (e.g. 16 x 1/4). Immediate feedback is provided.

Decimals Workshop - Online

Description: This innovative program allows students to perform decimals calculations in addition, subtraction, multiplication, and division. The program is totally customizable and allows users to select the number of problems and the numbers of digits before or after the decimal in each problem. It also provides a drag and drop, decimal-friendly work space

CC Standards: 5.NBT.A.3, 5.NBT.B.7, 6.NS.B.3

Multiplying a Whole Number by a Decimal (to the tenth) - Online

Description: This activity requires students to multiply a whole number by a tenth. For example, 8 x 0.4. Immediate feedback is given.

Multiplying Decimals to the Tenths - Online

Description: This activity requires students to multiply decimals to the tenths. For example, 4.3 x 2.7. Immediate feedback is given.

Multiplying Decimals to the Hundredths - Online

Description: This activity requires students to multiply decimals to the hundredths. For example, 2.35 x 4.72. Immediate feedback is given.

Multiplying Decimals to the Tenths and Hundredths - Online

Description: This activity requires students to multiply decimals to the tenths and hundredths. For example, 7.56 x 3.3. Immediate feedback is given.

Multiplying Three-digit by Three-digit Word Problems - Online

Description: This activity requires students to solve word problems with triple-digit multiplication. It gives immediate feedback.

Description: This activity requires students to identify the incorrect multiplication equations amongst eight world cup teams. The team with the least amount of incorrect equations wins! What country will win?

Format: Printable Activity

World Cup Multiplication - Online

Factorization Forest Practice (Version 1) - Online

## Group

• Closure:(a*b) belongs to G for all a,b &in G.
• Associativity: a*(b*c) = (a*b)*c ∀ a,b,c belongs to G.
• Identity Element:There exists e &in G such that a*e = e*a = a ∀ a &in G
• Inverses:∀ a &in G there exists a -1 &in G such that a*a -1 = a -1 *a = e
1. A group is always a monoid, semigroup, and algebraic structure.
2. (Z,+) and Matrix multiplication is example of group.

## Multiplication and Division Strategies

When are students are working on memorizing multiplication and division facts, they still very much need strategies to help them get the answer. This printable intervention activity works for that.

On these mats, the students are prompted to solve a multiplication or division problem using four different strategies. Depending on the needs of your students, you have a few options for how you want the students to work with the mats.

Option #1 – Have the students use all four strategies to solve the problem.

Option #2 – Have the students choose the most efficient way (in their opinions) to solve the problem. For example, some problems can be solved efficiently with repeated addition (4 x 9 = 9 + 9 + 9 + 9) and some work better when using related facts (7 x 8 = 2 x 8 and 5 x 8).

Not sure of the different multiplication and division strategies? I will also link multiplication and division strategy cards that you can use to review these strategies and that the students can then use as references.