Is G Abelian if f is a Homomorphism?

[hsong9]

Homework Statement

If G is any group, define f:G -> G by f(g) = g^-1
show that G is abelian if and only if f is a homomorphism.

The Attempt at a Solution

Suppose G is abelian.
Let a,b in G.
f(ab) = (ab)^-1 = b^-1 a^-1. Since G is abelian, b^-1 a^-1 = a^-1 b^-1.
we need to show that f(ab) = f(a)f(b)
f(ab) = f(a)f(b) = a^-1 b^-1 by define f.
so f is a homomorphism.

Suppose f is a homomorphism.
Let a,b in G.
Since f is a homomorphism, f(ab) = f(a)f(b) = a^-1 b^-1.
By define f, f(ab) = (ab)^-1 = b^-1 a^-1
Assume that a^-1b^-1 = b^-1 a^-1
a a^-1 b^-1 = a b^-1 a^-1
b^-1 = a b^-1 a^-1
b b^-1 = ba b^-1 a^-1
e = ba b^-1 a^-1
a = ba b^-1 a^-1 a
ab = ba b^-1 b
ab=ba, so G is abelian.

Correct?

 

[sutupidmath]

Since f is a homomorphism, f(ab) = f(a)f(b) = a^-1 b^-1.
By define f, f(ab) = (ab)^-1 = b^-1 a^-1
Assume that a^-1b^-1 = b^-1 a^-1


Correct?

It looks good to me. Besides,here when you say Assume that a^-1b^-1 = b^-1 a^-1 you don't really need to say so, because this follows imediately by assuming that f is homomorphism.

For the first part, it is correct, however i would write it this way

f(ab)=(ab)^-1=b^-1a^-1=a^-1b^-1=f(a)f(b).

Cheers!

 

[hsong9]

Yes, I don't need to say "Assume", Thanks!