Proving Group Homomorphism Between c and dc:G1--->G3

[kathrynag]

Homework Statement

Let c:G1--->G2 and d:G2--->G3 be group homomorphisms. Prove that dc:G1--->G3 is a homomorphism. Prove that ker(c) is a subset of ker(dc).

Homework Equations

The Attempt at a Solution

If a,b are in G1, then c(ab)=c(a)c(b) in G2 and so d(c(ab))=d(c(a)c(b))=d(c(a))d(c(b)) in G3

ker c is defined as x in G1 such that c(x)=e
ker dc is defined as x in G1 such that dc(x)=e
Then I get stuck

 

[micromass]

Take x in ker(c), then c(x)=e. What happens if you compose both sides with d?

 

[kathrynag]

c(x)=e
dc(x)=de
e=de
e=d

 

[kathrynag]

I guess I do that and am unsure where that leads me

 

[kathrynag]

c(x)=e
d(c(x))=d(e)
d(e)=d(e)
Ahhh, so that shows it

 

[micromass]

Yes, It looks like you've got it!