proof that group homomorphisms preserve identity

Theorem.

A group homomorphism preserves identity elements.

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
Canonical name	ProofThatGroupHomomorphismsPreserveIdentity
Date of creation	2013-11-16 4:44:43
Last modified on	2013-11-16 4:44:43
Owner	jacou (1000048)
Last modified by	(0)
Numerical id	9
Author	jacou (0)
Entry type	Proof
Classification	msc 20A05
Synonym	1234
Related topic	1234
Defines	1234