# 2: Groups I - Mathematics

We give a precise definition of a group and explore some different groups in the context of this definition.

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

## Math4Littles | Early Math Activities for Two- and Three-Year-Olds

We’ve designed these games to focus on the six key skill areas of early math.

When young children learn early math skills, it isn’t about equations and flashcards—it’s all about having fun while helping your little one’s brain grow. Take some time to browse the play activities below and try some with your 2-to-3 year-old. We’ve designed these games to focus on the six key skill areas of early math:

• Counting
• Computation
• Shapes
• Spatial awareness
• Measurement
• Patterns

Start with the first set of activities and then move on to the others when your child is ready. As you play, remember that children master skills at different speeds—for example, counting errors are common in the early years. Feel free to adjust the challenge level to suit your child. Remember the goal is having fun, so avoid making a big deal about mistakes. Just explain the correct answer and move on with the activity.

If you are looking for the Spanish translations of the activities, click here.

If you are a professional and would like to use these activities with the families in your program, check out the User’s Guide to assist with your planning and implementation.

Math4Littles is a collaboration between American Institutes of Research and ZERO TO THREE.

Bridges Intervention provides targeted instruction and assessment for essential K–5 mathematics skills within a tiered system of support. The small-group instruction and ongoing progress monitoring are consistent with a Response to Intervention (RTI) or Multi-Tiered System of Support (MTSS) framework.

Intended to complement regular math instruction, Bridges Intervention is ideal for small groups and can also be used with individuals. Students work with models that spur thinking and build confidence—starting with manipulatives, moving to two-dimensional representations and then mental images. Organized by content rather than grade, progress monitoring is key to the program. Each focused, 30-minute session is matched to student needs.

## Equal Groups

This means that you have two groups of 3 !

Put the two groups together . How many triangles do you have?

Count them. One, two, three, four, five, six!

Let's do another one! This one has the numbers switched around.

This means that you have three groups of 2 !

Put the three groups together . How many squares do you have?

Count them. One, two, three, four, five, six!

Hey, that's the same answer we got with 2 x 3 ! But, we put the six together a different way. Look at them both again to see the difference!

## Contents

• Cpt: Is this group Gcompact? (Yes or No)
• π 0 > : Gives the group of components of G. The order of the component group gives the number of connected components. The group is connected if and only if the component group is trivial (denoted by 0).
• π 1 > : Gives the fundamental group of G whenever G is connected. The group is simply connected if and only if the fundamental group is trivial (denoted by 0).
• UC: If G is not simply connected, gives the universal cover of G.
Lie group Description Cpt π 0 > π 1 > UC Remarks Lie algebra dim/R
R n Euclidean space with addition N 0 0 abelian R n n
R × nonzero real numbers with multiplication N Z2 abelian R 1
R + positive real numbers with multiplication N 0 0 abelian R 1
S 1 = U(1) the circle group: complex numbers of absolute value 1 with multiplication Y 0 Z R abelian, isomorphic to SO(2), Spin(2), and R/Z R 1
Aff(1) invertible affine transformations from R to R. N Z2 0 solvable, semidirect product of R + and R × < [ a b 0 0 ] : a , b ∈ R >a&b&0end> ight]:a,bin mathbb ight>> 2
H × non-zero quaternions with multiplication N 0 0 H 4
S 3 = Sp(1) quaternions of absolute value 1 with multiplication topologically a 3-sphere Y 0 0 isomorphic to SU(2) and to Spin(3) double cover of SO(3) Im(H) 3
GL(n,R) general linear group: invertible n×n real matrices N Z2 M(n,R) n 2
GL + (n,R) n×n real matrices with positive determinant N 0 Z n=2
Z2 n>2
GL + (1,R) is isomorphic to R + and is simply connected M(n,R) n 2
SL(n,R) special linear group: real matrices with determinant 1 N 0 Z n=2
Z2 n>2
SL(1,R) is a single point and therefore compact and simply connected sl(n,R) n 2 −1
SL(2,R) Orientation-preserving isometries of the Poincaré half-plane, isomorphic to SU(1,1), isomorphic to Sp(2,R). N 0 Z The universal cover has no finite-dimensional faithful representations. sl(2,R) 3
O(n) orthogonal group: real orthogonal matrices Y Z2 The symmetry group of the sphere (n=3) or hypersphere. so(n) n(n−1)/2
SO(n) special orthogonal group: real orthogonal matrices with determinant 1 Y 0 Z n=2
Z2 n>2
Spin(n)
n>2
SO(1) is a single point and SO(2) is isomorphic to the circle group, SO(3) is the rotation group of the sphere. so(n) n(n−1)/2
Spin(n) spin group: double cover of SO(n) Y 0 n>1 0 n>2 Spin(1) is isomorphic to Z2 and not connected Spin(2) is isomorphic to the circle group and not simply connected so(n) n(n−1)/2
Sp(2n,R) symplectic group: real symplectic matrices N 0 Z sp(2n,R) n(2n+1)
Sp(n) compact symplectic group: quaternionic n×n unitary matrices Y 0 0 sp(n) n(2n+1)
Mp(2n,R) metaplectic group: double cover of real symplectic group Sp(2n,R) Y 0 Z Mp(2,R) is a Lie group that is not algebraic sp(2n,R) n(2n+1)
U(n) unitary group: complex n×n unitary matrices Y 0 Z R×SU(n) For n=1: isomorphic to S 1 . Note: this is not a complex Lie group/algebra u(n) n 2
SU(n) special unitary group: complex n×n unitary matrices with determinant 1 Y 0 0 Note: this is not a complex Lie group/algebra su(n) n 2 −1

with Lie bracket the cross product also isomorphic to su(2) and to so(3,R)

The dimensions given are dimensions over C. Note that every complex Lie group/algebra can also be viewed as a real Lie group/algebra of twice the dimension.

The dimensions given are dimensions over C. Note that every complex Lie algebra can also be viewed as a real Lie algebra of twice the dimension.

where J is the standard skew-symmetric matrix

The Lie algebra of affine transformations of dimension two, in fact, exist for any field. An instance has already been listed in the first table for real Lie algebras.

## Categories

Below is a list of courses that tend to belong to each category. Some of these courses are not offered every year, new ones may be added, and our offering may change in other ways in any given year. For a current list of courses that count toward a particular category, with up-to-date prerequisites, we encourage you to use the attribute search on YCS.

### Algebra, Combinatorics, and Number Theory

(Math 350 and Math 370 are often taken as a 2-term sequence. Math 380 may also be taken for graduate credit. )

225 or 226 Linear Algebra

350 Introduction to Abstract Algebra (also carries core area algebra attribute)

353 Introduction to Representation Theory (typically offered every other year)

360 Introduction to Lie Groups (typically offered every other year)

370 Fields and Galois Theory (also carries core area algebra attribute)

373 Algebraic number theory (typically offered every other year)

380 Modern Algebra (also carries core area algebra attribute)

440 Introduction to Algebraic Geometry (typically offered every other year)

### Logic and Foundations

Phil 267 Mathematical L ogic (may count for pure math major only, with limit as noted above)

Phil 427 Computability and Logic (may count for pure math major only, with limit as noted above)

### Analysis

(Math 320-325 and Math 310-315 are generally taken as two term sequences Math 315, 320, and 325 may also be taken for graduate credit.)

246 Ordinary Differential Equations

302 Multivariable Analysis (also carries core area real analysis attribute)

305 Real Analysis (also carries core area real analysis attribute)

310 Introduction to Complex Analysis (also carries core area complex analysis attribute)

315 Intermediate Complex Analysis (also carries core area complex analysis attribute)

320 Measure Theory and Integration (also carries core area real analysis attribute)

325 Introduction to Functional Analysis (also carries core area real analysis attribute)

447 Partial differential equations (typically offered every other year)

### Geometry and Topology

360 Introduction to Lie Groups

430 Introduction to Algebraic Topology (typically offered every other year)

435 Differential Geometry (typically offered every other year)

544 Introduction to algebraic topology
(This is the only graduate course that carries an attribute.)

### Applied Mathematics

246 Ordinary Differential Equations

247 Partial Differential Equations

421 Mathematics of Data Science (typically offered every other year)

447 Partial differential equations (typically offered every other year)

### Other courses that may be of interest

As noted above, these only count for pure math majors (not joint-math), and there is a maximum of two that may be counted. They carry no attributes.

## A proof that the square root of 2 is irrational

Let's suppose &radic 2 is a rational number. Then we can write it &radic 2 = a/b where a, b are whole numbers, b not zero.

We additionally assume that this a/b is simplified to lowest terms, since that can obviously be done with any fraction. Notice that in order for a/b to be in simplest terms, both of a and b cannot be even. One or both must be odd. Otherwise, we could simplify a/b further.

From the equality &radic 2 = a/b it follows that 2 = a 2 /b 2 , or a 2 = 2 · b 2 . So the square of a is an even number since it is two times something.

From this we know that a itself is also an even number. Why? Because it can't be odd if a itself was odd, then a · a would be odd too. Odd number times odd number is always odd. Check it if you don't believe me!

Okay, if a itself is an even number, then a is 2 times some other whole number. In symbols, a = 2k where k is this other number. We don't need to know what k is it won't matter. Soon comes the contradiction.

If we substitute a = 2k into the original equation 2 = a 2 /b 2 , this is what we get:

This means that b 2 is even, from which follows again that b itself is even. And that is a contradiction.

WHY is that a contradiction? Because we started the whole process assuming that a/b was simplified to lowest terms, and now it turns out that a and b both would be even. We ended at a contradiction thus our original assumption (that &radic 2 is rational) is not correct. Therefore &radic 2 cannot be rational.

Irrational Numbers Rational Square Roots
How can you tell whether root 10 is a terminating or repeating decimal, or an irrational number? Are some square roots rational?

## Types of transformations in math

A translation or slide is an isometry in which all points of a figure move the same distance and in the same direction.

A reflection is an isometry in which the preimage and the image have opposite orientations. In other words, the image appears backwards.

A rotation is an isometry in which a figure has been rotated or turned around a point called center of rotation.

A dilation is a transformation whose preimage and image are similar. In general, a dilation is not an isometry. A dilation could be an enlargement or a reduction.

Composition of transformation

A composition of transformation is a combination of two or more transformations. For example, a figure could be translated and then reflected. A figure could be reflected and then rotated. A figure could be translated, reflected, rotated, and dilated. It all depends on the problem and the situation.

Looking at this figure again, can you tell what kind of transformations were performed on the preimage?

The preimage was moved to the right and we can see that the image is smaller than the preimage. Therefore, the preimage was translated and dilated.

## Are Boys Better Than Girls at Math?

Although the question of whether there is a gender difference in math seems like a simple one, the answer is complicated. Overall there are only small differences in boys&rsquo and girls&rsquo math performance those differences depend on the age and skill level of the student, what type of math they are attempting and how big of a dissimilarity is needed to say that boys&rsquo and girls&rsquo math performance is truly different.

In preschool and elementary school boys and girls generally perform similarly on math tests. Later in school, in high school and college, more consistent differences start to emerge. In addition, gender differences are often larger among higher-performing students but not necessarily for lower- or average-performing ones. Within this specific group of higher-performing math students, boys tend to perform better. Similarly, when studies do find gender differences among elementary school children, they find these start to appear for higher-performing students earlier in schooling than they do for lower- and average-performing ones.

Whether a gender difference is found also depends on what type of math the kids are doing. In general, boys tend to outperform girls on tests that are less related to what is taught in schools (like the SAT math test, for example) whereas there tend to be minimal gender differences on statewide standards-based math tests, which are more tied to what&rsquos taught in schools. When it comes to grades in school, which are even more closely tied to the curriculum, girls often outperform boys. A recent meta-analysis of research on the performance of students from elementary age through adulthood found boys tend to outperform girls in more complex areas of math such as those involving more advanced problem-solving. In contrast, there are no differences&mdashand, in some cases, an advantage for girls&mdashon more basic numerical skills and on math problems that have a set procedure for solving them.

Two of the factors above, age and the type of math, can impact research results at the same time. For example, two recent studies (here and here) found no gender differences in basic numerical skills in infants and children. This could be partially explained by the young age of the sample, and also because there are often few gender differences found in basic numerical skills.

Although there are differences in math performance between girls and boys of both high school and college age, and when doing certain typesof math, these studies find only a small gender difference in math performance. The mean performance scores for boys and girls are about 0.1 to 0.3 standard deviations apart from one another&mdashvery small differences and with a lot of overlap between boys&rsquo and girls&rsquo math skills. (Check out this visualization to see two groups that show a 0.2 standard deviation difference.) Thus, boys and girls are much more similar than different in math performance, even when considering studies that found the largest gender differences. In addition, even when we find there are differences, it is important to remember they are in the averages of the two groups and are not deterministic of any individual student&rsquos performance.

Interestingly, we often see larger gender difference in other math-related outcomes compared with overall performance. Girls tend to have less positive math attitudes: They have higher levels of math anxiety and lower levels of confidence in their math skills. This means even when girls show similar performance levels to boys, they are often less sure of themselves. In addition, we see larger gender differences in spatial skills, the way students approach solving math problems and math-intensive career choices. Considered along with the larger gender differences seen among higher-performing math students, who are the most likely to pursue a math-intensive career, varying spatial skills and problem-solving approaches, among other factors, may help us understand why boys go on to pursue math-intensive career choices more frequently than girls do. Therefore, these math-related skills and attitudes may be more useful areas for researchers to investigate related to gender and math.

Colleen Ganley is an assistant professor of developmental psychology and in the Florida Center for Research in STEM in the Learning Systems Institute at Florida State University. Her research focuses on the social, cognitive and attitudinal factors related to math learning with a specific interest in gender and income-level differences.

## 2: Groups I - Mathematics

Natasha Glydon

Consider this scenario: your school is planning to make toques and mitts to sell at the winter festival as a fundraiser. The school&rsquos sewing classes divide into two groups &ndash one group can make toques, the other group knows how to make mitts. The sewing teachers are also willing to help out. Considering the number of people available and time constraints due to classes, only 150 toques and 120 pairs of mitts can be made each week. Enough material is delivered to the school every Monday morning to make a total of 200 items per week. Because the material is being donated by community members, each toque sold makes a profit of $2 and each pair of mitts sold makes a profit of$5.

In order to make the most money from the fundraiser, how many of each item should be made each week? It is important to understand that profit (the amount of money made from the fundraiser) is equal to the revenue (the total amount of money made) minus the costs: Proft = Revenue - Cost. Because the students are donating their time and the community is donating the material, the cost of making the toques and mitts is zero. So in this case, profit &equiv revenue.

If the quantity you want to optimize (here, profit) and the constraint conditions (more on them later) are linear, then the problem can be solved using a special organization called linear programming. Linear programming enables industries and companies to find optimal solutions to economic decisions. Generally, this means maximizing profits and minimizing costs. Linear programming is most commonly seen in operations research because it provides a &ldquobest&rdquo solution, while considering all the constraints of the situation. Constraints are limitations, and may suggest, for example, how much of a certain item can be made or in how much time.

Creating equations, or inequalities, and graphing them can help solve simple linear programming problems, like the one above. We can assign variables to represent the information in the above problem.

x = the number of toques made weekly
y = the number of pairs of mitts made weekly

Then, we can write linear inequalities based on the constraints from the problem.

The students can only make up to 150 toques and up to 120 pairs of mitts each week. This is one restriction.

The total number of mitts and toques made each week cannot exceed 200. This is the material restriction.

We may also want to consider that x &ge 0 and y &ge 0. This means that we cannot make -3 toques.

Our final equation comes from the goal of the problem. We want to maximize the total profit from the toques and mitts. This can be represented by $2x +$5y = P, where P is the total profit, since there are no costs in production. If the school sells x toques, then they make $2x from the sales of toques. If the school sells y mitts, then they make$5y from the sales of mitts.

In some applications, the linear equations are very complex with numerous constraints and there are too many variables to work out manually, so they have special computers and software to perform the calculations efficiently. Sometimes, linear programming problems can be solved using matrices or by using an elimination or substitution method, which are common strategies for solving systems of linear equations.

Using the equations and inequations generated above, we can graph these, to find a feasible region. Our feasible region is the convex polygon that satisfies all of the constraints. In this situation, one of the vertices of this polygon will provide the optimal choice, so we must first consider all of the corner points of the polygon and find which pair of coordinates makes us the most money. From our toque and mitt example, we can produce the following graph:

We can see that our feasible region (the green area) has vertices of (0, 120), (150, 0),
(150, 50), and (80, 120). By substituting these values for x and y in our revenue equation, we can find the optimal solution.

R = 2x + 5y
R = 2(80) + 5(120)
R = \$760

After considering all of the options, we can conclude that this is our maximum revenue. Therefore, the sewing students (and teachers) must make 80 toques and 120 pairs of mitts each week in order to make the most money. We can check that these solutions satisfy all of our restrictions:
80 + 120 &le 200. This is true. We know that we will have enough material to make 80 toques and 120 pairs of mitts each week. We can also see that our values for x and y are less than 150 and 120, respectively. So, not only is our solution possible, but it is the best combination to optimize profits for the school. This is a fairly simple problem, but it is easy to see how this type of organization can be useful and very practical in the industrial world.

The airline industry uses linear programming to optimize profits and minimize expenses in their business. Initially, airlines charged the same price for any seat on the aircraft. In order to make money, they decided to charge different fares for different seats and promoted different prices depending on how early you bought your ticket. This required some linear programming. Airlines needed to consider how many people would be willing to pay a higher price for a ticket if they were able to book their flight at the last minute and have substantial flexibility in their schedule and flight times. The airline also needed to know how many people would only purchase a low price ticket, without an in-flight meal. Through linear programming, airlines were able to find the optimal breakdown of how many tickets to sell at which price, including various prices in between.

Airlines also need to consider plane routes, pilot schedules, direct and in-direct flights, and layovers. There are certain standards that require pilots to sleep for so many hours and to have so many days rest before flying. Airlines want to maximize the amount of time that their pilots are in the air, as well. Pilots have certain specializations, as not all pilots are able to fly the same planes, so this also becomes a factor. The most controllable factor an airline has is its pilot&rsquos salary, so it is important that airlines use their optimization teams to keep this expense as low as possible. Because all of these constraints must be considered when making economic decisions about the airline, linear programming becomes a crucial job.

• The military
• Capital budgeting
• Designing diets
• Conservation of resources
• Economic growth prediction
• Transportation systems (busses, trains, etc.)
• Strategic games (e.g. chess)
• Factory manufacturing

All of these industries rely on the intricate mathematics of linear programming. Even farmers use linear programming to increase the revenue of their operations, like what to grow, how much of it, and what to use it for. Amusement parks use linear programming to make decisions about queue lines. Linear programming is an important part of operations research and continues to make the world more economically efficient.

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.