Abstract Algebra/Group Theory/Homomorphism/Homomorphism Maps Inverse to Inverse
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< Abstract Algebra  Group Theory  Homomorphism
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, e_{K} is identity of K 5. e_{K} is identity of K