proof that group homomorphisms preserve identity


Let ϕ:GK be a group homomorphism. For clarity we use and for the group operationsMathworldPlanetmath of G and K, respectively. Also, denote the identitiesPlanetmathPlanetmathPlanetmath by 1G and 1H respectively.

By the definition of identity,

1G1G=1G. (1)

Applying the homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ϕ to (1) produces:

ϕ(1G)ϕ(1G)=ϕ(1G1G)=ϕ(1G). (2)

Multiply both sides of (2) by the inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmath of ϕ(1G) in K, and use the associativity of to produce:

ϕ(1G)=(ϕ(1G))-1ϕ(1G)ϕ(1G)=(ϕ(1G))-1ϕ(1G)=1K. (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