# How to prove that $G/\langle M\rangle/\langle N\rangle\cong G/\langle M\cup N\rangle$?

Suppose $(G,+,0)$ is an abelian group and $M,N\subseteq G$ two subsets.

Let $\langle M\rangle$ denotes the subgroup of $G$ generated by $M$ and let $\varphi:G\to G/\langle M\rangle$ denotes the quotient map.

Are $(G/\langle M\rangle)/\langle \varphi(N)\rangle$ and $G/\langle M\cup N\rangle$ isomorphic?

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The given overgroup $\,G\,$ is abelian so all its subgroups are normal... –  DonAntonio Oct 17 '12 at 18:37
You have obvious homomorphisms $[g]\to[g]$ in both directions, whose composition is of course the identity.