Skip to main content
\(\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{{\not{\mid}}} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Section3.2Definitions and Examples

The integers mod \(n\) and the symmetries of a triangle or a rectangle are examples of groups. A binary operation or law of composition on a set \(G\) is a function \(G \times G \rightarrow G\) that assigns to each pair \((a,b) \in G \times G\) a unique element \(a \circ b\text{,}\) or \(ab\) in \(G\text{,}\) called the composition of \(a\) and \(b\text{.}\) A group \((G, \circ )\) is a set \(G\) together with a law of composition \((a,b) \mapsto a \circ b\) that satisfies the following axioms.

  • The law of composition is associative. That is,

    \begin{equation*} (a \circ b) \circ c = a \circ (b \circ c) \end{equation*}

    for \(a, b, c \in G\text{.}\)

  • There exists an element \(e \in G\text{,}\) called the identity element, such that for any element \(a \in G\)

    \begin{equation*} e \circ a = a \circ e = a. \end{equation*}
  • For each element \(a \in G\text{,}\) there exists an inverse element in G, denoted by \(a^{-1}\text{,}\) such that

    \begin{equation*} a \circ a^{-1} = a^{-1} \circ a = e. \end{equation*}

A group \(G\) with the property that \(a \circ b = b \circ a\) for all \(a, b \in G\) is called abelian or commutative. Groups not satisfying this property are said to be nonabelian or noncommutative.

Most of the time we will write \(ab\) instead of \(a \circ b\text{;}\) however, if the group already has a natural operation such as addition in the integers, we will use that operation. That is, if we are adding two integers, we still write \(m + n\text{,}\) \(-n\) for the inverse, and 0 for the identity as usual. We also write \(m - n\) instead of \(m + (-n)\text{.}\)

It is often convenient to describe a group in terms of an addition or multiplication table. Such a table is called a Cayley table.

\begin{equation*} \begin{array}{c|ccccc} + & 0 & 1 & 2 & 3 & 4 \\ \hline 0 & 0 & 1 & 2 & 3 & 4 \\ 1 & 1 & 2 & 3 & 4 & 0 \\ 2 & 2 & 3 & 4 & 0 & 1 \\ 3 & 3 & 4 & 0 & 1 & 2 \\ 4 & 4 & 0 & 1 & 2 & 3 \end{array} \end{equation*}
Figure3.2.3Cayley table for \(({\mathbb Z_5}, +)\)

\begin{equation*} \begin{array}{c|cccc} \cdot & 1 & 3 & 5 & 7 \\ \hline 1 & 1 & 3 & 5 & 7 \\ 3 & 3 & 1 & 7 & 5 \\ 5 & 5 & 7 & 1 & 3 \\ 7 & 7 & 5 & 3 & 1 \end{array} \end{equation*}
Figure3.2.5Multiplication table for \(U(8)\)

A group is finite, or has finite order, if it contains a finite number of elements; otherwise, the group is said to be infinite or to have infinite order. The order of a finite group is the number of elements that it contains. If \(G\) is a group containing \(n\) elements, we write \(|G| = n\text{.}\) The group \({\mathbb Z}_5\) is a finite group of order 5; the integers \({\mathbb Z}\) form an infinite group under addition, and we sometimes write \(|{\mathbb Z}| = \infty\text{.}\)

Subsection3.2.1Basic Properties of Groups

Inverses in a group are also unique. If \(g'\) and \(g''\) are both inverses of an element \(g\) in a group \(G\text{,}\) then \(gg' = g'g = e\) and \(gg'' = g''g = e\text{.}\) We want to show that \(g' = g''\text{,}\) but \(g' = g'e = g'(gg'') = (g'g)g'' = eg'' = g''\text{.}\) We summarize this fact in the following proposition.

It makes sense to write equations with group elements and group operations. If \(a\) and \(b\) are two elements in a group \(G\text{,}\) does there exist an element \(x \in G\) such that \(ax = b\text{?}\) If such an \(x\) does exist, is it unique? The following proposition answers both of these questions positively.

This proposition tells us that the right and left cancellation laws are true in groups. We leave the proof as an exercise.

We can use exponential notation for groups just as we do in ordinary algebra. If \(G\) is a group and \(g \in G\text{,}\) then we define \(g^0 = e\text{.}\) For \(n \in {\mathbb N}\text{,}\) we define

\begin{equation*} g^n = \underbrace{g \cdot g \cdots g}_{n \; \text{times}} \end{equation*}

and

\begin{equation*} g^{-n} = \underbrace{g^{-1} \cdot g^{-1} \cdots g^{-1}}_{n \; \text{times}}. \end{equation*}

We will leave the proof of this theorem as an exercise. Notice that \((gh)^n \neq g^nh^n\) in general, since the group may not be abelian. If the group is \({\mathbb Z}\) or \({\mathbb Z}_n\text{,}\) we write the group operation additively and the exponential operation multiplicatively; that is, we write \(ng\) instead of \(g^n\text{.}\) The laws of exponents now become

  1. \(mg + ng = (m+n)g\) for all \(m, n \in {\mathbb Z}\text{;}\)

  2. \(m(ng) = (mn)g\) for all \(m, n \in {\mathbb Z}\text{;}\)

  3. \(m(g + h) = mg + mh\) for all \(n \in {\mathbb Z}\text{.}\)

It is important to realize that the last statement can be made only because \({\mathbb Z}\) and \({\mathbb Z}_n\) are commutative groups.

Subsection3.2.2Historical Note

Although the first clear axiomatic definition of a group was not given until the late 1800s, group-theoretic methods had been employed before this time in the development of many areas of mathematics, including geometry and the theory of algebraic equations.

Joseph-Louis Lagrange used group-theoretic methods in a 1770–1771 memoir to study methods of solving polynomial equations. Later, Évariste Galois (1811–1832) succeeded in developing the mathematics necessary to determine exactly which polynomial equations could be solved in terms of the polynomials'coefficients. Galois' primary tool was group theory.

The study of geometry was revolutionized in 1872 when Felix Klein proposed that geometric spaces should be studied by examining those properties that are invariant under a transformation of the space. Sophus Lie, a contemporary of Klein, used group theory to study solutions of partial differential equations. One of the first modern treatments of group theory appeared in William Burnside's The Theory of Groups of Finite Order [1], first published in 1897.