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Please focus your efforts on 50, 53, 54, 55, 56, 57, 58, 59, 60, 61. We'll come back to 51 and 52 on Wednesday.
Please spend 2-3 hours working on the problems below. Ideally, I'd love to have you come ready to present at least 3 of them (randomly pick 3). On Monday, I plan to split the class up into groups of 2-3, have you as a group put up a solution to one of these problems, and then we'll spend the hour looking at your solutions.
The following problem again asks you to make sure you are comfortable with the definitions of a binary operation and a group.
Problem 47 (Can We Use Division To Create A Group)
Let $G=\mathbb{R}$ and $H=\mathbb{R}\setminus \{0\}$.
- Show that division $a\div b$ is not a binary operationon $G$.
Definition (Binary Operation)
Let $G$ be a set. A binary operation on $G$ is a way of combining two elements of $G$ to obtain a new element in $G$. Formally, we just say that a binary operation $*$ is function $*:G\times G\to G$, and we use the notation $a*b$ to represent the function notation $*(a,b)$.
- Show that division $a\div b$ is a binary operation on $H$.
- Since division is a binary operation on $H$, determine if $(H,\div)$ 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.
- $\textbf{[Associativity]}$ For all $x,y,z\in G$ we have $(x* y)* z = x* (y* z)$.
- $\textbf{[Identity]}$ There is an $e\in G$ such that for all $x\in G$ we have $x * e = e* x = x$.
- $\textbf{[Inverses]}$ 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).
- Does $e=1$ satisfy the property of being an identity?
- If $x\in H$, find an inverse $x^{-1}\in H$ or explain why none exists.
- Is the operation $\div$ associative?
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.
- $\textbf{[Associativity]}$ For all $x,y,z\in G$ we have $(x* y)* z = x* (y* z)$.
- $\textbf{[Identity]}$ There is an $e\in G$ such that for all $x\in G$ we have $x * e = e* x = x$.
- $\textbf{[Inverses]}$ 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:
- $\textbf{[Closure]}$ for all $h,k\in H$ we have $h\cdot k\in H$, and
- $(H,\cdot)$ 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)
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.
- $\textbf{[Associativity]}$ For all $x,y,z\in G$ we have $(x* y)* z = x* (y* z)$.
- $\textbf{[Identity]}$ There is an $e\in G$ such that for all $x\in G$ we have $x * e = e* x = x$.
- $\textbf{[Inverses]}$ 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).
- If $a,b\in H$, then $ab\in H$. (We say $H$ is closed under the group operation.)
- If $a\in H$, then $a^{-1}\in H$. (We say $H$ is closed under taking inverses.)
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:
- $\textbf{[Closure]}$ for all $h,k\in H$ we have $h\cdot k\in H$, and
- $(H,\cdot)$ 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 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 $a^k=a^{-1}$.
Problem 51 (Inverses In Groups)
Suppose that $G$ is a group with $a,b\in G$.
- Prove that the inverse of $a^{-1}$ is $a$.
- Prove that the inverse of $ab$ is $b^{-1}a^{-1}$.
- 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.
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 $G$.
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 $$Z(G)=\{x\in G\mid gx=xg \text{ for all } g\in G\}.$$
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 $n\in \mathbb{N}$.
For more problems, see AllProblems