## 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 .

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.