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Introduction to Groups and Geometries

David W. Lyons

Section 2.2 Definition of a group

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 \(x,y,z\in S\text{.}\)

Definition 2.2.1. 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 \(g\ast h=h \ast g=e\text{.}\)

Definition 2.2.3. 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 .

Definition 2.2.4. Abelian group, additive notation.

In general, group operations are not commutative.
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Recall that a binary operation \(\ast\) on a set \(S\) is called commutative if \(x \ast y = y\ast x\) for all \(x,y\in S\text{.}\)
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 .

Definition 2.2.5. 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 \(G\text{.}\)

Definition 2.2.6. 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.

Exercises Exercises

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. Suppose that a group \(G\) is finite, and cyclic, with generator \(g\text{.}\) Show that
    \begin{equation*} G=\{g,g^2,g^3,\ldots,g^{|G|}\}. \end{equation*}
  2. The group of integers is cyclic. Find all of the generators.
  3. The group \(\Z_8\) is cyclic. Find all of the generators.
  4. The group \(\Z_2\times \Z_3\) is cyclic. Find all of the generators.
  5. Show that the group \(\Z_2\times \Z_2\) is not cyclic.
Hint.
For part a, let \(n\) be the least positive integer such that \(g^n=e\) (explain why \(n\) exists!). Given an arbitrary element \(h\in G\text{,}\) write \(h=g^k\) for some \(k\text{,}\) then use the Division Algorithm.

7. Cyclic permutations.

Let \(n\) be a positive integer and let \(k\) be an integer in the range \(2\leq k\leq n\text{.}\) 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. A \(1\)-cycle is defined to be the identity permutation, and may be written as \((a)\) in cycle notation, for any \(a\in \{1,2,\ldots,n\}\text{.}\) Note that cycle notation is not unique. For example, in \(S_4\) we have \((2,4,3)=(4,3,2)=(3,2,4)\) and \((1)=(2)=(3)=(4)\text{.}\) A permutation is called cyclic if it is a \(k\)-cycle for some \(k\text{,}\) \(1\leq k \leq n\text{.}\)
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There are two conventions about whether the identity permutation is considered cyclic. According to the definition in this text, the identity is cyclic because it is a 1-cycle. Another convention, not used here, is that a permutation is cyclic if it is a \(k\)-cycle for some \(k\geq 2\text{.}\) According to this convention, the identity is not cyclic.
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 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. Every permutation is a product of transpositions.

  1. Show that every permutation in \(S_n\) can be written as a product of transpositions.
  2. Show that factoring a permutation into a product of transpositions is not unique by writing the identity permutation in \(S_3\) as a product of transpositions in two different ways.

10. 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.
\begin{equation*} \begin{array} {c|cccccc} \amp e \amp (23) \amp (13) \amp (12) \amp (123) \amp (132) \\ \hline e \amp \amp \amp \amp (12) \amp \amp \\ (23) \amp \amp \amp \amp \amp \amp \\ (13) \amp \amp (132) \amp \amp \amp \amp \\ (12) \amp \amp \amp \amp \amp (23) \amp \\ (123) \amp \amp \amp \amp \amp \amp \\ (132) \amp \amp \amp \amp \amp \amp \end{array} \end{equation*}
Figure 2.2.7. (Partial) Cayley table for \(S_3\text{.}\) The symbol \(e\) denotes the identity permutation.
\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 \amp R_{1/2} \amp \amp \amp \amp \amp \amp \\ F_H \amp \amp \amp \amp \amp F_D \amp \amp \amp \\ F_D \amp \amp \amp \amp \amp \amp F_{D'} \amp \amp \\ F_{D'} \amp \amp \amp \amp \amp \amp \amp \amp \\ R_{1/4}\amp \amp \amp \amp \amp \amp \amp \amp \\ R_{1/2} \amp \amp \amp \amp \amp \amp \amp \amp \\ R_{3/4} \amp \amp \amp \amp \amp \amp \amp \amp \\ R_0\amp \amp \amp \amp \amp \amp \amp \amp \end{array} \end{equation*}
Figure 2.2.8. (Partial) Cayley table for \(D_4\text{.}\) (See Checkpoint 2.1.6 for notation for the elements of \(D_4\text{.}\))
  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*}
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