The discussion focuses on proving two properties of group homomorphisms: that the image of the identity element under a homomorphism is the identity element of the target group, expressed as f(e) = e', and that the image of the inverse of an element is the inverse of the image, expressed as f(g^-1) = f(g)^-1. The first property is established through the definition of the neutral element, while the second property remains unresolved, prompting further inquiry into the relationship between f(gg^-1) and the homomorphic property f(x·y) = f(x)·f(y).
PREREQUISITESStudents of abstract algebra, mathematicians focusing on group theory, and anyone seeking to understand the fundamental properties of homomorphisms in algebraic structures.
LagrangeEuler said:Homework Statement
Show that
##f(e)=e'## and ##f(g^-1)=f(g)^-1##
Homework Equations
Homomorphism
f(x\cdot y)=f(x)\cdot f(y)
The Attempt at a Solution
I show the first one. Neutral element is element which satisfied
##e\cdot e=e##.
So
##f(e)=f(e\cdot e=e)=f(e)\cdot f(e)##
So ##f(e)=e'##.
But how to show
##f(g^{-1})=f(g)^{-1}##?