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

Proposition 2.2.2. Immediate consequences of the definition of group.

Let \(G\) be a group. The element e in the second property of Definition 2.2.1 is unique. Given \(g\in G\text{,}\) the element h in the third property of Definition 2.2.1 is unique.

Proof.

See Exercise 2.2.2.1 and Exercise 2.2.2.2.

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.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 .

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

Exercise 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 \(e=e'\text{.}\)

Exercise 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 \(h=h'\text{.}\)

Exercise 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 \(g=h\text{.}\)

Exercise 4

4. The "socks and shoes" property.

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

Exercise 5

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

\[
(g,h)\ast_{G\times H} (g',h')= (g\ast_G g',h\ast_H
h').
\nonumber \]

Show that this operation satisfies the definition of a group.

Exercise 6

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

\[
\left(g^k\right)_{k\in
\mathbb{Z}}=(\ldots,g^{-3},g^{-2},g^{-1},g^0,g^1,g^2,g^3,\ldots)
\nonumber \]

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

1. The group of integers is cyclic. Find all of the generators.
2. The group \(\mathbb{Z}_8\) is cyclic. Find all of the generators.
3. The group \(\mathbb{Z}_2\times \mathbb{Z}_3\) is cyclic. Find all of the generators.
4. Show that the group \(\mathbb{Z}_2\times \mathbb{Z}_2\) is not cyclic.
5. Let m,n be positive integers. Show that the group \(\mathbb{Z}_m\times \mathbb{Z}_n\) 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 \((a,b)\in \mathbb{Z}_m\times \mathbb{Z}_n\) is a generator if and only if every entry in the sequence

\[
(a,b),(2a,2b),(3a,3b),\ldots,(mna,mnb)
\nonumber \]

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.

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

\[
a_1{\to} a_2{\to} a_3{\to} \cdots{\to} a_k{\to} a_1
\nonumber \]

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

\[
2\to 4\to 3\to 2
\nonumber \]

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 \(S_4\text{.}\)

Exercise 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.

Exercise 9

Show that every permutation in \(S_n\) can be written as a product of transpositions.

Exercise 10

Parity of a permutation.

1. Suppose that the identity permutation \(e\) in \(S_n\) is written as a product of transpositions

   \[
   e=\tau_1\tau_2\cdots \tau_r.
   \nonumber \]

   Show that \(r\) is even.
2. Suppose that \(\sigma\) in \(S_n\) is written in two ways as a product of transpositions.

   \[
   \sigma = (a_1b_1)(a_2b_2)\cdots (a_sb_s) =
   (c_1d_1)(c_2d_2)\cdots (c_td_t)
   \nonumber \]

   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

a. 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.

\[
(ab)(ab), (ac)(ab), (bc)(ab), (cd)(ab)
\nonumber \]

The first allows you to reduce the transposition count by two by cancelling. The remaining three can be rewritten.

\[
(ab)(bc), (ac)(cb), (ab)(cd)
\nonumber \]

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.

Exercise 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.

\[
\begin{array}
{c|cccccc}
& e & (23) & (13) & (12) & (123) & (132) \\
\hline
e & & & & (12) & & \\
(23) & & & & & & \\
(13) & & (132) & & & & \\
(12) & & & & & (23) & \\
(123) & & & & & & \\
(132) & & & & & &
\end{array}
\nonumber \]

Figure 2.2.7. (Partial) Cayley table for \(S_3\text{.}\) The symbol e denotes the identity permutation.

\[
\begin{array}
{c|cccccccc}
& F_V & F_H & F_D & F_{D'} & R_{1/4} & R_{1/2} & R_{3/4} &
R_0\\ \hline
F_V & & R_{1/2} & & & & & & \\
F_H & & & & & F_D & & & \\
F_D & & & & & & F_{D'} & & \\
F_{D'} & & & & & & & & \\
R_{1/4} & & & & & & & & \\
R_{1/2} & & & & & & & & \\
R_{3/4} & & & & & & & & \\
R_0 & & & & & & & &
\end{array}
\nonumber \]

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

\[
\begin{array}
{c|cccccc}
& e & (23) & (13) & (12) & (123) & (132) \\
\hline
e & e & (23) & (13) & (12) & (123) & (132) \\
(23) & (23) & e & (123) & (132) & (13) & (12)\\
(13) & (13) & (132) & e & (123) & (12) & (23)\\
(12) & (12) & (123) & (132) & e & (23) & (13)\\
(123) & (123) & (12) & (23) & (13) & (132) & e\\
(132) & (132) & (13) & (12) & (23) & e & (123)
\end{array}
\nonumber \]

Answer 2

\[
\begin{array}
{c|cccccccc}
& F_V & F_H & F_D & F_{D'} & R_{1/4} & R_{1/2} & R_{3/4} &
R_0\\ \hline
F_V & R_0 & R_{1/2} & R_{3/4} & R_{1/4} & F_{D'} & F_H & F_D & F_V \\
F_H & R_{1/2} & R_0 & R_{1/4} & R_{3/4} & F_D & F_V & F_{D'} & F_H \\
F_D & R_{1/4} & R_{3/4} & R_0 & R_{1/2} & F_V & F_{D'} & F_H & F_D \\
F_{D'} & R_{3/4} & R_{1/4} & R_{1/2} & R_0 & F_H & F_D & F_V & F_{D'} \\
R_{1/4} & F_D & F_{D'} & F_H & F_V & R_{1/2} & R_{3/4} & R_0 & R_{1/4} \\
R_{1/2} & F_H & F_V & F_{D'} & F_D & R_{3/4} & R_0 & R_{1/4} & R_{1/2} \\
R_{3/4} & F_{D'} & F_D & F_V & F_H & R_0 & R_{1/4} & R_{1/2} & R_{3/4} \\
R_0 & F_V & F_H & F_D & F_{D'} & R_{1/4} & R_{1/2} & R_{3/4} & R_0
\end{array}
\nonumber \]