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Section 2.2 Definition of a group

Subsection 2.2.1

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

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. 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 \(x,y\in S\text{.}\)
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 2.2.2 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. 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. Find necessary and sufficient conditions on positive integers \(n,m\) such that the group \(\Z_n\times \Z_m\) is cyclic, and prove your statement.
Hint
For the last part, observe that \((a,b)\in \Z_n\times \Z_m\) is a generator if and only if every entry in the sequence
\begin{equation*} (a,b),(2a,2b),(3a,3b),\ldots,(nma,nmb) \end{equation*}
is distinct. Consider \((La,Lb)\text{,}\) where \(L\) is the least common multiple of \(n,m\text{.}\)
7. Cyclic permutations.

A permutation \(\pi\) of a set \(X\) is called a full cycle if the sequence

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

contains all the elements of \(X\) for some \(x\) in \(X\text{.}\) A permutation \(\pi\) of a set \(X\) is called a \(k\)-cycle if there is a \(k\)-element set \(A\subseteq X\) such that \(\pi\) acts as a full cycle on \(A\) and \(\pi\) acts as the identity on the complement of \(A\) (this means that the sequence \(\left(\pi^k(a)\right)_{k\in \Z}\) contains all the elements of \(A\) for some \(a\) in \(A\) and that \(\pi(x)=x\) for all \(x\) in \(X\setminus A\)). For example, the element \(\pi=[1,4,2,3]\) 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. In \(S_n\text{,}\) we write \((a_1a_2\cdots a_k)\) 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 set \(A=\{a_1,a_2,\ldots,a_k\}\subseteq \{1,2,\ldots,n\}\text{,}\) where \(a_1,a_2,\ldots,a_k\) are distinct integers. For example, we have \([1,4,2,3]=(2,4,3)\) in \(S_4\text{.}\) Note that this 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.

\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*}