what is an automorphism of G?

1. it is a homomorphism from G to G

2. it is bijective

so we have to check conditions (1) and (2) above.

to prove (1), we need to show that for all x,y in G.

but can you continue?

to prove (2), we have to show 2 things as well:

2.a. is injective

2.b. is surjective

to prove (2.a) it suffices to show that . so suppose .

this means that , so , so what can we say about x? (what is the unique element of G such that ax = a?).

to prove (2.b) for any y in G, we need to find some x in G with . how about ?

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alternately, to show is bijective, we could exhibit a (two-sided) inverse. what is the logical candidate for ?

(maybe ? can you prove this ?)