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I have some difficulties with the following problem:

Let $G$ be a group with a composition serie $\mathcal G$; Let $A$ and $B$ be maximal normal subgroups of $G$; if $A'$ is a maximal normal subgroup of $A$ and $B'$ is a maximal normal subgroup of $B$ prove that $\left$ is a subnormal subgroup of $G$.

The situation is the following:

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Which is the strategy to approach this problem (it is enough only a hint for the moment)? I'm looking for a solution for several days... :(

Edited: I should prove the porposition in the case "$G$ is a group (infinite order is allowed) with a composition serie... ". There is no need that $G$ is finite.

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    What I said is also still true, no need for finiteness. Just an induction on the length of the series.2012-10-12

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Hint: I think this approach would work: try to prove that

  1. the lattice of subnormal subgroups is modular: $ U\subset W \implies (U,V\cap W) = (U,V)\cap W $
  2. the desired property holds for modular lattices

Update: ..hmmm.. I might have misunderstood the question. So, we don't know yet that $\langle A',B'\rangle$ belongs to this lattice...