# abstract algebra – Second Isomorphism Theorem and Jordan-Holder

In another posting, there was a question about the following:

Let $$G$$ be a finite non-trivial group with the following two composition series:

$${e} = M_0 triangleleft M_1 triangleleft M_2 = G$$

$${e} = N_0 triangleleft N_1 triangleleft cdots triangleleft N_r = G.$$

Prove that $$r=2$$ and that $$G/M_1 cong G/N_1$$ and $$N_1/N_0 cong M_1/M_0$$.

The person posting the question went on to state that “By the second isomorphism theorem I know that $$M_1N_2/N_2 cong M_1/(N_2cap M_1)$$

Here is my question. In order to use the second isomorphism theorem with $$M_1$$ and $$N_2$$ don’t we need to know that $$M_1 leq N_G(N_2)$$? And if so, then how do we know that $$M_1$$ actually is in the normalizer of $$N_2$$ in $$G$$?

If I can get over this hurdle, I understand the remainder of the original posting. Perhaps this is something obvious, but please help me see whatever it is that I am missing.