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(d).
Since c and d are homomorphisms, we know c(ab)=c(a)c(b) and d(ab)=d(a)d(b).
We want to show dc(ab)=dc(a)dc(b)
I get stuck at this point.
Well, first, trying to say doesn't make any sense; the two are not even contained in the same group.
I believe the question should be to show that
Now, just think about it; what would it mean to say ?
Purely by definition of what it means to be a subset, it means that we want to show that, if , then as well. Why should this be true?