Let us compare the group
under addition modulo
with the cyclic subgroup of generated by :

We see that the two tables differ only in the names of the symbols, and not in their positions. More formally, there is mapping (namely , , ) which is a bijection and satisfies .

It makes sense to regard isomorphic groups as identical. The main general task of group theory can be formulated as: classify all non-isomorphic groups. In general this is impossible, and one has to settle for various partial results in this direction. Probably the easiest such is the following:

In order to prove that two groups and are not isomorphic, one needs to demonstrate that there is no isomorphism from onto . Usually, in practice, this is much easier than it sounds in general, and is accomplished by finding some property that holds in one group, but not in the other.

Finally, we prove the so called Cayley's theorem, which suggests a prominent role of the symmetric groups among all groups.

The practical use of Cayley's Theorem is limited: it is not very likely that one can obtain much useful information about groups of order, say, eight, by considering subgroups of the group of order .

Edmund F Robertson

11 September 2006