Please Login to access more options.


Today

« October 2013 »

Sun

Mon

Tue

Wed

Thu

Fri

Sat

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

 FrontRightBackLeft
1AustinLiBryceCassie
2CaleyCaleyTerryTerry
3MeganKinseyAlexanderJustin
4BrennanBrennanSamSam
5    
6KimberlyKimberlyNickNick
7xxxx
8JoeJoeLiliaLilia
9CarlaCarlaLauraLaura

For Friday, the problems below are to help you practice with the group axioms. Spend 15 minutes on each problem. Make sure you are ready to come to class to present a few of each, and hopefully all. Each problem has a fairly short answer.

In class, we'll split up into small groups and take turns presenting. If you have not presented in class very much, you need to make sure you are ready to present Friday.

Click to see the definitions of a group and subgroup, as well as the subgroup test.

Definition (Group)

Let $G$ be a nonempty set, and let $*$ be a binary operation on $G$, which means for every $x,y\in G$ we have $x*y\in G$ $\textbf{[Closure]}$. The structure $\mathbb{G} = (G,*)$ is called a $\textdef{group}$ if the following hold.

  1. $\textbf{[Associativity]}$ For all $x,y,z\in G$ we have $(x* y)* z = x* (y* z)$.
  2. $\textbf{[Identity]}$ There is an $e\in G$ such that for all $x\in G$ we have $x * e = e* x = x$.
  3. $\textbf{[Inverses]}$ For all $x\in G$ there is a $y\in G$ such that $x* y = y* x = e$.

We usually simply write $G$ when referring to the entire structure $\mathbb{G}=(G,*)$. The element $e$ from the second point is called the $\textdef{identity}$. The element $y$ from the third point is called the $\textdef{inverse}$ of $x$ and is usually denoted $x^{-1}$. One often simply writes $xy$ in place of $x*y$, and for every positive integer $n$, we'll write $x^n$ as shorthand for $x* x* \cdots * x$ ($n$ times).

Definition (Subgroup)

Let $(G,\cdot)$ be a group, and let $H$ be a nonempty subset of $G$. Then $H$ is called a $\textdef{subgroup}$ of $G$ if the following hold:

  1. $\textbf{[Closure]}$ for all $h,k\in H$ we have $h\cdot k\in H$, and
  2. $(H,\cdot)$ is a group.

When $H$ is a subgroup of $G$, we write $H\le G$. Any subgroup of $G$ that is not equal to $G$ itself we call a $\textdef{proper subgroup}$. The subset of $G$ consisting of just the identity we call the $\textdef{trivial subgroup}$.

Problem 46 (The Subgroup Test - Subgroups Are Subsets That Are Closed Under Products And Taking Inverses)

Suppose that $G$ is a
group
Definition (Group)

Let $G$ be a nonempty set, and let $*$ be a binary operation on $G$, which means for every $x,y\in G$ we have $x*y\in G$ $\textbf{[Closure]}$. The structure $\mathbb{G} = (G,*)$ is called a $\textdef{group}$ if the following hold.

  1. $\textbf{[Associativity]}$ For all $x,y,z\in G$ we have $(x* y)* z = x* (y* z)$.
  2. $\textbf{[Identity]}$ There is an $e\in G$ such that for all $x\in G$ we have $x * e = e* x = x$.
  3. $\textbf{[Inverses]}$ For all $x\in G$ there is a $y\in G$ such that $x* y = y* x = e$.

We usually simply write $G$ when referring to the entire structure $\mathbb{G}=(G,*)$. The element $e$ from the second point is called the $\textdef{identity}$. The element $y$ from the third point is called the $\textdef{inverse}$ of $x$ and is usually denoted $x^{-1}$. One often simply writes $xy$ in place of $x*y$, and for every positive integer $n$, we'll write $x^n$ as shorthand for $x* x* \cdots * x$ ($n$ times).

and $H$ is a nonempty subset of $G$ that satisfies the following properites:
  1. If $a,b\in H$, then $ab\in H$. (We say $H$ is closed under the group operation.)
  2. If $a\in H$, then $a^{-1}\in H$. (We say $H$ is closed under taking inverses.)
Prove that $H$ is a
subgroup
Definition (Subgroup)

Let $(G,\cdot)$ be a group, and let $H$ be a nonempty subset of $G$. Then $H$ is called a $\textdef{subgroup}$ of $G$ if the following hold:

  1. $\textbf{[Closure]}$ for all $h,k\in H$ we have $h\cdot k\in H$, and
  2. $(H,\cdot)$ is a group.

When $H$ is a subgroup of $G$, we write $H\le G$. Any subgroup of $G$ that is not equal to $G$ itself we call a $\textdef{proper subgroup}$. The subset of $G$ consisting of just the identity we call the $\textdef{trivial subgroup}$.

of $G$.

Problem 49 (Cancellation Laws For Groups)

Suppose $G$ is a group. Let $a,b,c\in G$. Prove that if $ca=cb$, then $a=b$. A similar proof will show that if $ac=bc$, then $a=b$.

Problem 50 (The Inverse In A Finite Group Is A Power Of The Element)

Let $G$ be finite group with $a\in G$. Prove that there exists a positive integer $k$ such that $a^k=a^{-1}$.

Hint: Consider the sequence $a, a^2, a^3, a^4, \ldots$. Why must you eventually have a repeated element? So if $a^i=a^j$ for some $j>i$, how can you use this to find $a^{-1}$?

Problem 51 (Inverses In Groups)

Suppose that $G$ is a group with $a,b\in G$.

  1. Prove that the inverse of $a^{-1}$ is $a$.
  2. Prove that the inverse of $ab$ is $b^{-1}a^{-1}$.
  3. If $a_1,a_2,a_3,\ldots, a_n\in G$, state the inverse of $a_1a_2a_3\cdots a_n$. Use induction to prove your claim.
If you see yourself repeating an induction proof similar to what we've been doing, then you're on the right track.

Problem 52 (Heisenberg Matrix Group)

Let $G$ be the set of all 3 by 3 matrices with entries in the real numbers of the form $$\begin{bmatrix}1&a&b\\0&1&c\\0&0&1\end{bmatrix}.$$ Prove that $G$ is group under matrix multiplication. This group is often called the Heisenberg group and is connected to the Heisenberg uncertainty principle. See page 51 in your text for an interesting historical fact.

Problem 53 (Finite Subgroup Test)

Let $G$ be a group. Suppose that $H$ is a nonempty finite subset of $G$ and that $H$ is closed under the operation of $G$ (so if $a,b\in H$, then we must have $ab\in H$). Prove that $H$ is a subgroup of $G$.

Hint. If you use the subgroup test (show that $H$ is a nonempty subset of $G$ that is closed under the operation and taking inverses), then we get to assume it's a nonempty subset of $G$ that is closed under the operation. All you have to do is explain why it's closed under taking inverses. The work you did in the problem The Inverse In A Finite Group Is A Power Of The Element should help you quite a bit. However, you can't use this theorem directly because you do not know that $G$ is a finite group. You'll want to reuse your work from that problem, not refer to the problem.

Problem 54 (The Intersection Of Two Subgroups)

Suppose that $G$ is a group and that $H$ and $K$ are subgroups of $G$. Prove that $H\cap K$ is a subgroup of $G$.

Problem 55 (The Union Of Two Subgroups)

Suppose that $G$ is a group and that $H$ and $K$ are subgroups of $G$. Is $H\cup K$ a subgroup of $G$? Either prove that is, or find a counterexample.

Definition (Abelian Group)

Let $G$ be a group. If $ab=ba$ for every $a,b\in G$ (so the group operation is commutative), then we say that $G$ is Abelian.

Definition ($Z(G)$ - Center Of A Group)

Let $G$ be a group. The center of the group, written $Z(G)$, is the set of elements $x\in G$ that commute with every element of $G$, which we can write symbolically as $$Z(G)=\{x\in G\mid gx=xg \text{ for all } g\in G\}.$$

The $Z$ comes from the german word "Zentrum" (see Wikipedia).

Problem 56 (The Center Of Group Is A Subgroup)

Prove that the center $Z(G)$ of a group $G$ is a subgroup of $G$. If $G$ is Abelian, then what is $Z(G)$?

Problem 57 (Powers Of Products In An Abelian Group)

Suppose $G$ is an Abelian group. Prove that if $a,b\in G$, then $(ab)^2=a^2b^2$. Then use induction to prove that if $a,b\in G$, then $(ab)^n=a^nb^n$ for each $n\in \mathbb{N}$.

On Monday, we'll be returning to the problems from Wednesday, so feel free to head back and look at these problems again.

For more problems, see AllProblems