Have you ever thought you knew something well and talked about it for years, only to suddenly find out one day you were wrong? (Shaking my head in despair!) I thought I knew without a shadow of a doubt what an isomorphism was...

The problem: Let G be a group and Aut G the set of all automorphisms of G.

a) Show Aut G is a group. (Easy. I did this.)

b) Show

and

.

Part b) is what's bothering me. Let's take the second case first since it typifies the problem I'm having.

The automorphism is the set of all bijective maps

. Now, we may think of the general element of this set essentially as an element of the set of permutations of {0,1,2,3,4,5}. This means the set

has 6! elements. The set

has 2 elements.