Prove c is an isomorphism iff G is abelian
Let G be any group. Define c:G-->G by c(x)=x^-1 for all x in G. prove that c is an isomorphism iff G is abelian.
Let c be an isomorphism. We want to show G is abelian.
Since c is an isomorphism, c is 1-1, onto, and a homomorphism.
We have c(a)=c(b) implies a=b by 1-1
We have c(x)=y by onto
G must be abelian since (ab)^-1=b^-1a^-1
Now suppose G is abelian. We want to show this implies c is an isomorphism.
Since G is abelian, we know ab=ba
1-1 and onto follow by the same ideas as before
Since c is abelian (ab)^-1=a^-1b^-1
I am really not sure if I did any of those correct