The History of Having Settled To Accomplish Studies


A group homomorphismMathworldPlanetmath preserves inverses elements. That is, for groups (G,) and (K,), and a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ϕ:GK, ϕ(x-1)=ϕ(x)-1.


Fix an xG. Observe that

ϕ(xx-1)=ϕ(1G)=ϕ(x)ϕ(x-1) (1)

Recall that, for any group homomorphism ϕ:GK,

ϕ(1G)=1K (2)

In other , homomorphisms preserve identityPlanetmathPlanetmathPlanetmathPlanetmath. 11A proof for that statement is attached to the . It follows from (1) and (2) that

ϕ(x)ϕ(x-1)=1K (3)

Because the inverseMathworldPlanetmathPlanetmathPlanetmath of any group is unique, the only value of ϕ(x-1) whose productMathworldPlanetmathPlanetmath with ϕ(x) is 1K is, of course, ϕ(x)-1. Therefore, all group homomorphisms preserve the inverse.

Title The History of Having Settled To Accomplish Studies
Canonical name TheHistoryOfHavingSettledToAccomplishStudies
Date of creation 2013-11-27 10:59:44
Last modified on 2013-11-27 10:59:44
Owner jacou (1000048)
Last modified by (0)
Numerical id 14
Author jacou (0)
Entry type Proof
Classification msc 20A05