proof that group homomorphisms preserve identity

Theorem.

A group homomorphism preserves identity elements.

Proof.

Let $\varphi :G\to K$ 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 $\varphi$ to (1) produces:

$$\varphi (1_G)\star \varphi (1_G)=\varphi (1_G\ast 1_G)=\varphi (1_G).$$(2)

Multiply both sides of (2) by the inverse of $\varphi (1_G)$ in $K$,and use the associativity of $\star$ to produce:

$$\varphi (1_G)={(\varphi (1_G))}^{-1}\star \varphi (1_G)\star \varphi (1_G)={(\varphi (1_G))}^{-1}\star \varphi (1_G)=1_K.$$(3)

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