Abstract Algebra/Group Theory/Homomorphism/Homomorphism Maps Inverse to Inverse

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Theorem[edit]

Let f be a homomorphism from group G to Group K.

Let g be any element of G.

f(g-1) = [f(g)]-1

Proof[edit]

0.   f is a homomorphism
1.   definition of inverse in G
.
2.   homomorphism f maps identity to identity
3.   as f(g) is in K, so is its inverse [f(g)]-1
.
4.   inverse on K, eK is identity of K
5.   eK is identity of K