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## Section2.2Definition of a group

### Subsection2.2.1

We will use the notation $\ast \colon S\times S\to S$ to denote a binary operation on a set $S$ that sends the pair $(x,y)$ to $x\ast y\text{.}$ Recall that a binary operation $\ast$ is associative means that $x\ast(y\ast z)= (x\ast y)\ast z$ for all $x,y,z\in S\text{.}$

###### Definition2.2.1.Group.

A group is a set $G\text{,}$ together with a binary operation $\ast\colon G\times G \to G$ with the following properties.

• The operation $\ast$ is associative.
• There exists an element $e$ in $G\text{,}$ called an identity element, such that $e\ast g=g\ast e=g$ for all $g\in G\text{.}$
• For every $g\in G\text{,}$ there exists an element $h\in G\text{,}$ called an inverse element for $g\text{,}$ such that $g\ast h=h \ast g=e\text{.}$
###### Definition2.2.3.Multiplicative notation.

Let $G$ be a group. By Proposition 2.2.2, we may speak of an identity element as the identity element for $G\text{.}$ Given $g\in G\text{,}$ we may refer to an inverse element for $g$ as the inverse of $g\text{,}$ and we write $g^{-1}$ to denote this element. In practice, we often omit the operator $\ast\text{,}$ and simply write $gh$ to denote $g\ast h\text{.}$ We adopt the convention that $g^0$ is the identity element. For $k\geq 1\text{,}$ we write $g^k$ to denote $\underbrace{g\ast g\ast \cdots \ast g}_{k \text{ factors}}$ and we write $g^{-k}$ to denote $\left(g^{k}\right)^{-1}\text{.}$ This set of notational conventions is called multiplicative notation .

###### Definition2.2.4.Abelian group, additive notation.

In general, group operations are not commutative. 1  A group with a commutative operation is called Abelian.

For some Abelian groups, such as the group of integers, the group operation is called addition, and we write $a+b$ instead of using the multiplicative notation $a\ast b\text{.}$ We write $0$ to denote the identity element, we write $-a$ to denote the inverse of $a\text{,}$ and we write $ka$ to denote $\underbrace{a+ a+ \cdots +a}_{k \text{ summands}}$ for positive integers $k\text{.}$ This set of notational conventions is called additive notation .

Recall that a binary operation $\ast$ on a set $S$ is called commutative if $x \ast y = y\ast x$ for all $x,y\in S\text{.}$
###### Definition2.2.5.Order of a group.
The number of elements in a finite group is called the order of the group. A group with infinitely many elements is said to be of infinite order. We write $|G|$ to denote the order of the group $G\text{.}$
###### Definition2.2.6.The trivial group.

A group with a single element (which is necessarily the identity element) is called a trivial group. In multiplicative notation, one might write $\{1\}\text{,}$ and in additive notation, one might write $\{0\}\text{,}$ to denote a trivial group.

### Exercises2.2.2Exercises

###### 1.Uniqueness of the identity element.

Let $G$ be a group. Suppose that $e,e'$ both satisfy the second property of the Definition 2.2.1, that is, suppose $e\ast x=x\ast e = e'\ast x=x\ast e'=x$ for all $x\in G\text{.}$ Show that $e=e'\text{.}$

###### 2.Uniqueness of inverse elements.

Let $G$ be a group with identity element $e\text{.}$ Let $g\in G$ and suppose that $g\ast h = h\ast g = g\ast h' = h'\ast g = e\text{.}$ Show that $h=h'\text{.}$

###### 3.The cancellation law.

Suppose that $gx=hx$ for some elements $g,h,x$ in a group $G\text{.}$ Show that $g=h\text{.}$ [Note that the same proof, mutatis mutandis, shows that if $xg=xh\text{,}$ then $g=h\text{.}$]

###### 4.The "socks and shoes" property.

Let $g,h$ be elements of a group $G\text{.}$ Show that $(gh)^{-1} = h^{-1}g^{-1}\text{.}$

###### 5.Product Groups.

Given two groups $G,H$ with group operations $\ast_G,\ast_H\text{,}$ the Cartesian product $G\times H$ is a group with the operation $\ast_{G\times H}$ given by

\begin{equation*} (g,h)\ast_{G\times H} (g',h')= (g\ast_G g',h\ast_H h'). \end{equation*}

Show that this operation satisfies the definition of a group.

###### 6.Cyclic groups.

A group $G$ is called cyclic if there exists an element $g$ in $G\text{,}$ called a generator, such that the sequence

\begin{equation*} \left(g^k\right)_{k\in \Z}=(\ldots,g^{-3},g^{-2},g^{-1},g^0,g^1,g^2,g^3,\ldots) \end{equation*}

contains all of the elements in $G\text{.}$

1. The group of integers is cyclic. Find all of the generators.
2. The group $\Z_8$ is cyclic. Find all of the generators.
3. The group $\Z_2\times \Z_3$ is cyclic. Find all of the generators.
4. Show that the group $\Z_2\times \Z_2$ is not cyclic.
5. Let $m,n$ be positive integers. Show that the group $\Z_m\times \Z_n$ is cyclic if and only if $m,n$ are relatively prime, that is, if the greatest common divisor of $m,n$ is 1.
Hint
For the last part, observe that $(a,b)\in \Z_m\times \Z_n$ is a generator if and only if every entry in the sequence
\begin{equation*} (a,b),(2a,2b),(3a,3b),\ldots,(mna,mnb) \end{equation*}
is distinct (say why!). Let $L$ be the least common multiple of $n,m\text{.}$ If $m,n$ are relatively prime, then $L=mn\text{,}$ and if $m,n$ are not relatively prime, then $L\lt mn$ (say why!). Use this observation to prove the statement in the exercise.
###### 7.Cyclic permutations.

Let $n$ be a positive integer and $k$ be an integer in the range $1\leq k\leq n\text{.}$ A permutation $\pi\in S_n$ (see Definition 2.1.1) is called a $k$-cycle if there is a $k$-element set $A=\{a_1,a_2,\ldots,a_k\}\subseteq \{1,2,\ldots,n\}$ such that $\pi(a_i)=a_{i+1}$ for $1\leq i\leq k-1$ and $\pi(a_k)=a_1\text{,}$ and $\pi(j)=j$ for $j\not\in A\text{.}$ We use cycle notation $(a_1a_2\cdots a_k)$ to denote the $k$-cycle that acts as

\begin{equation*} a_1{\to} a_2{\to} a_3{\to} \cdots{\to} a_k{\to} a_1 \end{equation*}

on the distinct positive integers $a_1,a_2,\ldots,a_k\text{.}$ For example, the element $\pi=[1,4,2,3]=(2,4,3)$ is a 3-cycle in $S_4$ because $\pi$ acts on the set $A=\{2,3,4\}$ by

\begin{equation*} 2\to 4\to 3\to 2 \end{equation*}

and $\pi$ acts on $A^c=\{1\}$ as the identity. Note cycle notation is not unique. For example, we have $(2,4,3)=(4,3,2)=(3,2,4)$ in $S_4.$ Cycles of any length (any positive integer) are called cyclic permutations. A 2-cycle is called a transposition.

1. Find all of the cyclic permutations in $S_3\text{.}$ Find their inverses.
2. Find all of the cyclic permutations in $S_4\text{.}$
###### 8.

Cycles $(a_1a_2\cdots a_k)$ and $(b_1b_2\cdots b_\ell)$ are called disjoint if the the sets $\{a_1,a_2,\ldots,a_k\}$ and $\{b_1,b_2,\ldots,b_\ell\}$ are disjoint, that is, if $a_i\neq b_j$ for all $i,j\text{.}$ Show that every permutation in $S_n$ is a product of disjoint cycles.

###### 9.

Show that every permutation in $S_n$ can be written as a product of transpositions.

###### 10.Parity of a permutation.
1. Suppose that the identity permutation $e$ in $S_n$ is written as a product of transpositions
\begin{equation*} e=\tau_1\tau_2\cdots \tau_r. \end{equation*}
Show that $r$ is even.
2. Suppose that $\sigma$ in $S_n$ is written in two ways as a product of transpositions.
\begin{equation*} \sigma = (a_1b_1)(a_2b_2)\cdots (a_sb_s) = (c_1d_1)(c_2d_2)\cdots (c_td_t) \end{equation*}
Show that $s,t$ are either both even or both odd. The common evenness or oddness of $s,t$ is called the parity of the permutation $\sigma\text{.}$
3. Show that the parity of a $k$-cycle is even if $k$ is odd, and the parity of a $k$-cycle is odd if $k$ is even.
Hint
1. Consider the two rightmost transpositions $\tau_{r-1}\tau_{r}\text{.}$ They have one of the following forms, where $a,b,c,d$ are distinct.
\begin{equation*} (ab)(ab), (ac)(ab), (bc)(ab), (cd)(ab) \end{equation*}
The first allows you to reduce the transposition count by two by cancelling. The remaining three can be rewritten.
\begin{equation*} (ab)(bc), (ac)(cb), (ab)(cd) \end{equation*}
Notice that the index of the rightmost transposition in which the symbol $a$ occurs has been reduced by 1 (from $r$ to $r-1$). Finish this reasoning with an inductive argument.
###### 11.Cayley tables.

The Cayley table for a finite group $G$ is a two-dimensional array with rows and columns labeled by the elements of the group, and with entry $gh$ in position with row label $g$ and column label $h\text{.}$ Partial Cayley tables for $S_3$ (Figure 2.2.7) and $D_4$ (Figure 2.2.8) are given below.

1. Fill in the remaining entries in the Cayley tables for $S_3$ and $D_4\text{.}$
2. Prove that the Cayley table for any group is a Latin square. This means that every element of the group appears exactly once in each row and in each column.
Answer 1
\begin{equation*} \begin{array} {c|cccccc} \amp e \amp (23) \amp (13) \amp (12) \amp (123) \amp (132) \\ \hline e \amp e \amp (23) \amp (13) \amp (12) \amp (123) \amp (132) \\ (23) \amp (23) \amp e \amp (123) \amp (132) \amp (13) \amp (12)\\ (13) \amp (13) \amp (132) \amp e \amp (123) \amp (12) \amp (23)\\ (12) \amp (12) \amp (123) \amp (132) \amp e \amp (23) \amp (13)\\ (123) \amp (123) \amp (12) \amp (23) \amp (13) \amp (132) \amp e\\ (132) \amp (132) \amp (13) \amp (12) \amp (23) \amp e \amp (123) \end{array} \end{equation*}
Answer 2
\begin{equation*} \begin{array} {c|cccccccc} \amp F_V \amp F_H \amp F_D \amp F_{D'} \amp R_{1/4} \amp R_{1/2} \amp R_{3/4} \amp R_0\\ \hline F_V \amp R_0 \amp R_{1/2} \amp R_{3/4} \amp R_{1/4} \amp F_{D'} \amp F_H \amp F_D \amp F_V \\ F_H \amp R_{1/2} \amp R_0 \amp R_{1/4} \amp R_{3/4} \amp F_D \amp F_V \amp F_{D'} \amp F_H \\ F_D \amp R_{1/4} \amp R_{3/4} \amp R_0 \amp R_{1/2} \amp F_V \amp F_{D'} \amp F_H \amp F_D \\ F_{D'} \amp R_{3/4} \amp R_{1/4} \amp R_{1/2} \amp R_0 \amp F_H \amp F_D \amp F_V \amp F_{D'} \\ R_{1/4} \amp F_D \amp F_{D'} \amp F_H \amp F_V \amp R_{1/2} \amp R_{3/4} \amp R_0 \amp R_{1/4} \\ R_{1/2} \amp F_H \amp F_V \amp F_{D'} \amp F_D \amp R_{3/4} \amp R_0 \amp R_{1/4} \amp R_{1/2} \\ R_{3/4} \amp F_{D'} \amp F_D \amp F_V \amp F_H \amp R_0 \amp R_{1/4} \amp R_{1/2} \amp R_{3/4} \\ R_0 \amp F_V \amp F_H \amp F_D \amp F_{D'} \amp R_{1/4} \amp R_{1/2} \amp R_{3/4} \amp R_0 \end{array} \end{equation*}