Expert: David Hemmer - 3/13/2007

Question
Hi,
Can you please help me in proving the follwoing:

prove that isomorphism is an equivelance realtion on the family of groups.

Thanks

Answer
Suppose G is a group. The map i(g)=g is an isomorphism from G to itself so the relation is reflexive.

Suppose f:G->H is an isomorphism. Then in particular f is a bijection so it has an inverse map, f^{-1} which is also a group homorphism, and hence and isomorphism from H to G. Thus the relation is symmetric.

Finally suppose f1:G->H and f2:H->K are isomorphisms. The map f2 composed with f1 is an isomorphism from G to K so the relation is transitive.