Let $M$, $N$ be normal subgroups of a group $G$ such that $G/M$ and $G/N$ are solvable groups. How can I prove that $G/(M\cap N)$ and $G/\langle M,N\rangle$ are solvable either?
Thanks in advance.
2026-04-04 07:21:29.1775287289
Solvability of the groups.
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Hints:
$$MN/M\cong N/\left(M\cap N\right)$$
$$G/ N\cong\left(G/\left(M\cap N\right)\right)/\left(N/\left(M\cap N\right)\right)$$
$$G/MN\cong\left(G/N\right)/\left(MN/N\right)$$