411 REFERENCE · Wikibooks
Abstract Algebra/Group Theory/Homomorphism/Homomorphism Maps Inverse to Inverse
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Theorem
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
{| Style = "width:75%;boarder:1px"
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| 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
|}