# proof that group homomorphisms preserve identity

###### Proof.

Let $\phi:G\to K$ be a group homomorphism. For clarity we use $\ast$ and $\star$ for the group operations of $G$ and $K$, respectively. Also, denote the identities by $1_{G}$ and $1_{H}$ respectively.

By the definition of identity,

 $1_{G}\ast 1_{G}=1_{G}.$ (1)

Applying the homomorphism $\phi$ to (1) produces:

 $\phi(1_{G})\star\phi(1_{G})=\phi(1_{G}\ast 1_{G})=\phi(1_{G}).$ (2)

Multiply both sides of (2) by the inverse of $\phi(1_{G})$ in $K$, and use the associativity of $\star$ to produce:

 $\phi(1_{G})=(\phi(1_{G}))^{-1}\star\phi(1_{G})\star\phi(1_{G})=(\phi(1_{G}))^{% -1}\star\phi(1_{G})=1_{K}.$ (3)

Title proof that group homomorphisms preserve identity ProofThatGroupHomomorphismsPreserveIdentity 2013-11-16 4:44:43 2013-11-16 4:44:43 jacou (1000048) (0) 9 jacou (0) Proof msc 20A05 1234 1234 1234