Thread: Characteristic subgroups problem

Let N be a characteristic subgroup of a group G. Prove that, if and is a characteristic subgroup of , then K is a characteristic subgroup of G.

So I can find an automorphism f such that f(N) = N, and I will need to find another automorphism g such that g(K) = K.

So I know that I can find h such that h(k/N)=K/N.

But I'm having trouble trying to deal with the factor group, h would be a mapping from G/N onto itself, right? So how can I relate that to G itself?

Thank you!

Originally Posted by tttcomrader

Let N be a characteristic subgroup of a group G. Prove that, if and is a characteristic subgroup of , then K is a characteristic subgroup of G.

let define by: since is charateristic, is well-defined and injective. it follows that

note that since is characteristic and contained in we have: thus: which gives us: