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)

∎

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