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I'd love to have someone take pictures of what you put on the board (anyone want to volunteer), and then email those to me. I'll be back on Monday,
Br. Woodruff
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).
You've been studying groups since you started adding integers back in grade school. Just about every idea we've encountered since the start of the semester has also been a group. Now that we have isolated the key properties that make up a group, we need to become comfortable with showing that sets with a binary operation are groups, i.e. we need to become comfortable with checking the four properties of closure, associativity, identity, and inverses.
Let's practice showing that some sets are groups by showing that $\mathbb{Z}_n$ and $U(n)$ are groups. If we already showed a key fact in a previous problem, feel free to refer to problem by name and state the fact that was proved there. We have already shown most of the reasons why the sets below are groups.
Problem 44 ($\mathbb{Z}_n$ and $U(n)$ are groups)
Show the following. You need to briefly explain why the set together with its binary operation satisfies the definition of a group.
- For each $n\geq 1$, the set $\mathbb{Z}_n$ is a group under addition mod $n$.
- For each $n\geq 2$, prove that $U(n)$ is a group under multiplication mod $n$.
Try to solve the problems above without looking up the definition of a group.
If you need to, click here to show the definition of 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).
Because of how we defined groups, anytime we span a collection of permutations of a set $X$ we should get a group. The next problem has you show this. Each question can be answered by referring to things we have already shown. Feel free to look at the pages September 2016 and October 2016 for a list of all the problems we have solved each day. If you have not yet printed the work we are doing, I would suggest that you print a copy for reference as you work on new problems.
Problem 45 (Spans Of Permutations Are Subgroups)
Let $X=\{1,2,3,\ldots,n\}$. Let $S_n$ be the set of all permutations of $X$ (so we know there are $n!$ such permutations).
- Show that $S_n$ is a group under function composition $\circ$.
- If $H$ is a subset of $S_n$ that is closed (under composition combinations of permutations), prove that $H$ is a group under function composition.
- Let $S$ be a nonempty subset of $S_n$. Show that $\text{span}(S)$ is a group under function composition.
In the previous problem, we saw if $S\subseteq S_n$ then $\text{span}(S)$ is a not only just a subset of the group $S_n$, but is itself a group. The operation we use in the smaller group $(\text{span}(S),\circ)$ comes from the operation we used in the larger group $(S_n,\circ)$. Because of this close connection, we call $\text{span}(S)$ a subgroup of $S_n$ and write $\text{span}(S)\leq S_n$. Let's now make a formal definition.
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 $\textdef{trivial subgroup}$.
In the proof of problem Characterizing Closed Sets Of Permutations, we did not need to use the first and fourth properties listed to show that the set of permutations $H$ was closed. The next problem replicates this, but now in terms of subgroups instead of in terms of closed sets of permutations.
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 $\textdef{trivial subgroup}$.
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?
Definition ($|a|$ and $|G|$ - Order For Elements and Groups)
Let $G$ be a group with identity $e$, and let $g\in G$.
- The $\textdef{order}$ of $G$, denoted $|G|$, is the cardinality of $G$.
- The $\textdef{order}$ of $g$, denoted $|g|$, is the smallest positive integer $n$ such that $g^n = e$, if such an $n$ exists. If no such $n$ exists, we say $g$ has infinite order.
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