Department of Mathematics and Computer Science

Clark University

*G*is closed under composition: if*T*and*U*both belong to*G,*then so does the composition*T*°*U.*- The identity transformation
*I*belongs to*G.* *G*is closed under inversion: if*T*belongs to*G,*then so does*G*^{-1}.

We need some axioms for an abstract group. First of all, there needs to be some binary operation, an abstraction of composition. Since the elements of an abstract group don't have to actually be transformations, and the operation doesn't have to actually be composition, we'll call it something else. Usually it's called multiplication, but in some applications it might be addition or something else. We'll call it multiplication here.

Now, composition is always associative, so we'll include an axiom stating that our binary operation of multiplication is associative.

We also need an identity element (which will be the identity transformation when we have a transformation group), and we'll need inverses.

All these requirements boil down to the following formal definition.

- associativity: if
*x, y,*and*z*are three elements of*G*, then*x*(*yz*) = (*xy*)*z,* - identity: there is an element of
*G*called the*identity*element and denoted 1 (or sometimes something else like*e*) such that for each element*x*of*G*it is that case that*x*1 = 1*x*=*x,*and *G*is closed under inversion: for each element*x*of*G*there is an element of*G*called its*inverse*and denoted*x*^{-1}(or sometimes something else), such that*x*^{-1}*x*=*x x*^{-1}= 1.

** Theorem.** Each element of a group has exactly one inverse.
That is, for

**Some examples besides transformation groups.**
Although every group can be construed as a transformation group (we'll
see how later), many are as easily understood as abstract groups.
Here are a few basic examples.

The integers * Z.* An

There are a lot of other groups with addition as their binary
operations. For these groups the binary operation is commutative
as well as associative, that is,
*x* + *y* = *y* + *x*
for any two elements *x* and *y* of the group. When the
binary operation is commutative, we say the group is an *Abelian
group* in honor of Niels Henrik Abel (1802-1829). Such groups are
sometimes simply called *commutative groups.*

The general linear group GL_{n}(* R*). We saw that
the set of

Sometimes, groups are presented by "generators and relations." We'll look at that in detail later, but this is a good place to mention it. For an example, consider this presentation:

This says that two of the elements of *G* are *x* and *y.*
Other elements are formed by inverses and products of the "generating
elements." For instance,
*y*^{-2}*xy*^{3}*x*^{-1}
is one such other element. The presentation also states that the
square of *x* equals the cube of *y.* That implies that
the element just mentioned could also be written as
*y*^{-2}*x*^{2} since
*xy*^{3}*x*^{-1}
= *xx*^{2}*x*^{-1} = *x*^{2}.
One of the difficulties with groups presented by generators and
relations is it's hard to determine when two elements are equal.
For instance, with this one, can you tell if
*xy* equal to *yx*? (It's not.)

Copyright © 2003.
David E. Joyce

These pages are located on the web at

http://aleph0.clarku.edu/~djoyce/modalg/