Groups, An Introduction

Introduction

A group is a set of elements S and an operator * possessing the following properties:

1. The binary operator * is a well defined function mapping S cross S into S.

2. The operator * is associative.

3. There is a unique identity element 1, such that x*1 = 1*x = x for every x in S. This identity element is sometimes denoted e. When the group operator is commutative the identity element is often denoted 0, and * is replaced with +.

4. every x has a unique inverse y such that x*y = y*x = 1.

The group is abelian if * is commutative. Why don't they just call it a commutative group? It is called abelian in honor of Niels Abel. (biography)

As with regular multiplication, the star is often omited. Thus x*y is simply written xy, and x*x*x*x is written x⁴. Realize that x⁴yx is not x⁵y, since x and y may not commute.

If the group is abelian, we often use + instead of * and 0 instead of 1. This reminds us that the elements can be rearranged as we wish. For instance, x+x+y+x+y = 3x+2y.

The integers form an abelian group under addition, with 0 as the identity element. The positive rationals form an abelian group under multiplication, with 1 as the identity element.

The integers mod n, Z_n, form a group under addition, and the units mod n, denoted Z_n^*, form a group under multiplication.

Use the clock to illustrate addition mod 12, written Z12, which is an abelian group. Add 4 hours + 3 hours + 7 hours, in any order, and wind up at 2:00. And the inverse of 4 is 8, since 4+8 takes you back to the top. The set of 3 by 3 real orthonormal matrices form a nonabelian group under matrix multiplication. Thinking geometrically, this is the rigid rotations and reflections about the origin in 3 space. A rotation about the x axis and a rotation about the y axis do not commute, but a sequence of 3 rotations is always associative, and every rotation has an inverse. You can always spin the object back to start. Try it with a standard die. Spin it around x, then around y, and compare that with the rotation about y, then about x. The operations do not commute. A semigroup has no identity or inverses, such as the positive reals under addition. A monoid is a semigroup with an identity, such as the non-negative reals. Here 0 is the identity, but without negative numbers we have no inverses. The simplest monoid, that is not a group, consists of 0 and x, where x+x = x.