Homomorphism and Preimage: How are they related in Group Theory?

[kathrynag]

Homework Statement

Let i : G → H be a homomorphism of groups. Fix an element g of G and let
i(g) = h ∈ H. Show that the preimage i^−1(h) of h under i is the set
i^−1(h) = {kg | k ∈ ker i}.

Homework Equations

The Attempt at a Solution

i(ab)=i(a)i(b)
preimage is a in G such that i(a)=h
We know i(a)=h by definition of i
i(a)=h
i(a)i(a)^-1=hi(a)^-1
e=hi(a)^-1

 

[annoymage]

let x be the preimage of h, ie: i(x)=h=i(g),
then i(xg-1)=e, then what can you say about x??

 

[kathrynag]

It's the kernel?

 

[annoymage]

hmm, you know xg-1 in in the kernel, then (xg-1)g is in {kg | k ∈ ker i}, so this conclude that i^−1(h) is subset of {kg | k ∈ ker i}.
now left to show is {kg | k ∈ ker i} subset of i^−1(h)