# Subgroups of Quotient Groups

1. Feb 21, 2008

### Hello Kitty

Let G be a group and let $$N\trianglelefteq G$$, $$M\trianglelefteq G$$ be such that $$N \le M$$. I would like to know if, in general, we can identify $$G/M$$ with a subgroup of $$G/N$$.

Of course the obvious way to proceed is to look for a homomorphism from $$G$$ to $$G/N$$ whose kernel is M, but I can't think of one.

What I actually want to show is a more specialized result (namely the case when finite $$G/N$$ is the nilpotent quotient of $$G$$ and $$G/M$$ is a maximal p-quotient of $$G$$ for some p dividing the order of $$G/N$$) but the above is a lot cleaner and didn't yield obviously to a proof or counter-example so I thought I'd explore that first.