An Infinite non-nilpotent group whose every maximal subgroup is a normal subgroup.

abstract-algebragroup-theorynilpotent-groupsnormal-subgroups

It is widely known that for every finite group $G$, $G$ is nilpotent If and only if every maximal subgroup of $G$ is a normal subgroup.

But I don't know if there is an infinite non-nilpotent group whose every maximal subgroup is a normal subgroup.

Thank you for your help.

Best Answer

Yes, there is. There is an example of a non-nilpotent group with the normalizer condition, i.e., $H<N_G(H)$ for all proper subgroups $H$. It was constructed by Heineken and Mohamed in 1968. The paper is

A group with trivial centre satisfying the normalizer condition, J. Algebra 10, 368-376 (1968).

Edit: This group is seriously wonky. From their paper, it has:

  • $Z(G)=1$.
  • $G'$ is of exponent $p$ and abelian
  • Every proper subgroup of $G$ is subnormal and nilpotent.
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