It is well-known that a finite group is nilpotent if and only if every maximal subgroup is normal. Furthermore every nilpotent group has all maximal subgroups normal.

Question: Let $G$ be a group whose maximal subgroups are normal in $G$. If $M$ is a maximal subgroup of $G$, then are all the maximal subgroups of $M$ normal in $M$.

Of course for finite groups this is true since any subgroup of a nilpotent group is nilpotent, so my question is really about non-finite groups.

Edit 2015-11-05:

In order to get some idea of whether or not this claim is true it would be very helpful to look at some examples of non-nilpotent groups (admitting maximal subgroups) in which every maximal subgroup is normal. If someone knows of relevant literature it would be appreciated.

  • $\begingroup$ your question: let $G$ is a group with all maximal subgroups normal; if $M$ is a maximal subgroup, then are all maximal subgroups of $M$ normal (in $M$ or $G$?). Please clarify to me. $\endgroup$ – Groups Oct 28 '15 at 7:01
  • $\begingroup$ @Groups Thank you for your comment. I edited the question to make it clearer. $\endgroup$ – Nex Oct 28 '15 at 7:03
  • $\begingroup$ Sorry. Hypothesis is "$M$ is any maximal subgroup of $G$; all the maximal subgroups of ...... are normal in ......". Then is ........? (please, if possible, let me know these blanks; I want to know-I am not getting it.) $\endgroup$ – Groups Oct 28 '15 at 7:08
  • $\begingroup$ @Groups Is it clear now? $\endgroup$ – Nex Oct 28 '15 at 7:11
  • $\begingroup$ Great. Thats clear to me. Thanks very very much for effort. $\endgroup$ – Groups Oct 28 '15 at 7:14

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