Because I get bored with calculus and New Age thinking, I shall continue my quest to bring my (limited version of) abstract mathematics to this forum. And the first person to say "get a life" will have lost theirs (only teasing!). But groups are at one and the same intuititive and surprising. Let's see.

First let's have adefinition:

A group G is a set which has

a well-defined, associative and closed operation

an identity e

an inverse.

Let's unpack this. In group theory, the group operation is always referred to as "group multiplication". But don't be fooled: group theorists don't always mean multplication in the arithmetic sense; it might mean addition. There is a convention which I'll explain later.

The operation, group multiplication, is said to be "closed" if the result of the operation is still an element of the group (are the primes then a group?).

The identity, usually written e, is 0 when the operation is (arithmetic) addition, 1 when it is (arithmetic) multiplication. The inverses are respectively -x and 1/x.

Now some notational conventions. Some authors use the centre dot to denote the group operation thus aÂ·b, but most use juxtaposition thus ab. Don't assume this means arithmetic product, though, we might be in

an additive group.

Similarly most books use x<sup>-1</sup> for the inverse, to be interpreted according to the operation in question

With this in mind, we see that if a, b are in G, ab = c implies c is in G. We also see that ae = a , and aa<sup>-1</sup> = e. Note that these last equalities refer to what are called right identity and right inverse, respectively. The left identity and inverse follow follow from the group axioms. (Anybody want to try the simple proof?)

Now, perhaps the most important distinction we can make between different sorts of groups is this: if the operation on the elements h, g of a group G is commutative (i.e. gh = hg) the group is said to be abelian, otherwise non-abelian. I referred earlier to a notational convention: for abelian groups the operation is said to be addition.

Now non-commutivity need not throw us into a panic - think matrix multiplication for example, but the notation can be misleading at first. ab = c need not imply that ba = c, but it is always the case that ab = c means a = cb<sup>-1</sup> = b<sup>-1</sup>c, by my axioms above.

One more word on commutivity; it, or the lack of it, is a property of the group elements, not of the operation (associativity would not follow otherwise).

Roughly speaking, the order of a group G is the number of elements in the underlying set |G|. There's an interesting theorem of my buddy Lagrange, but first I need to tell you this.

Consider the groups G and H. If every element of H is also an element of G, clearly G = H iff every element of G is also an element of H. But if there are elements in G not in H we may say H is a subgroup of G. As H is a group in its own right it must share the identity e with G, must have an inverse and must be closed under the group operation. Evidently the operation on G and H must be the same.

OK. Lagrange says that the order of any subgroup H of G divides the order of G. This is less easy to prove than it looks, but it's a really cool result.

I'm going to close with some examples of groups and things which aren't groups:

Z, the integers, is an additive abelian group

The even integers are an additive abelian group; the odds are not a group, neither are the primes

S<sub>n</sub>, the set of permutations on n objects is a group, non-abelian for n > 2

The set of rotations in 3-space are a non-abelian group.

Tired of typing, more another time if anyone wants