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, Zn, form a group under addition, and the units mod n, denoted Zn*, 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.

rotations about the x and y axes do not commute