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

###### 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.

*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 .

*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

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

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

- The group of integers is cyclic. Find all of the generators.
- The group \(\Z_8\) is cyclic. Find all of the generators.
- The group \(\Z_2\times \Z_3\) is cyclic. Find all of the generators.
- Show that the group \(\Z_2\times \Z_2\) is
*not*cyclic. - 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.

###### 7. Cyclic permutations.

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

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

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

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.

- Find all of the cyclic permutations in \(S_3\text{.}\) Find their inverses.
- 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.

- 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.
- 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 \(\sigma\text{.}\)
- 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.

- 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.

- Fill in the remaining entries in the Cayley tables for \(S_3\) and \(D_4\text{.}\)
- 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.