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Definition: Let $G$ be a group, and let $H \leq G$ be a subgroup. We say that $H$ is big in $G$ if for every intermediate subgroup $H \leq L \leq G$ there exists some $x \in L$ such that $\langle H \cup \{x\} \rangle = L$.

Question: Is there some $n \in \mathbb{N}$ such that every big subgroup of $A_5^n$ surjects onto $A_5$?

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    $\begingroup$ I suspect ${A_4}^n$ is big in ${A_5}^n$, but do not know this. Gerhard "Not Fully Remembering Subdirectly Irreducible" Paseman, 2015.06.08 $\endgroup$
    – Gerhard Paseman
    Commented Jun 8, 2015 at 21:33
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    $\begingroup$ @GerhardPaseman Yes it's easy to check that if $G_i$ are finitely many groups and $M_i$ are non-normal maximal subgroups, then any subgroup of $\prod G_i$ containing $\prod M_i$, has the form $\prod H_i$, where $H_i\in\{M_i,G_i\}$ for all $i$. $\endgroup$
    – YCor
    Commented Jun 8, 2015 at 22:02
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    $\begingroup$ I think A_4^n being big means the answer to the question is no. $\endgroup$
    – The Masked Avenger
    Commented Jun 8, 2015 at 23:18

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