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Let's call "characteristically simple" such groups, that do not have proper non-trivial characteristic subgroups. Does there exist some sort of classification of finite "characteristically simple" groups?

All simple groups are "characteristically simple", but not all finite "characteristically simple" groups are simple. For example, if $G$ is a simple group and $n$ is a positive integer, then $G^n$ (a direct product of $n$ copies of $G$) is "characteristically simple", but, if $2 \leq n$, it is not simple. It will be interesting to know, whether there are "characteristically simple" groups not of that form.

Chain Markov
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1 Answers1

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It is well-known that all characteristically simple groups are of the form $G^n$ for some simple group $G$. This is important when you develop the basics of finite primitive groups, as the proof strategy of the O'Nan-Scott theorem is based on understanding the minimal normal subgroups in the group, which are characteristically simple.

A. Pongrácz
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