- #1
hsong9
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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?