Let $G$ be a group and $\mathfrak{L}(G)$ be set of all subgroups of $G$. Clearly, $\mathfrak{L} (G)$ is a lattice.

If we know that $\mathfrak{L} (G)$ is symmetric then what can be said about the group $G$ ?

Any reference and observation would be appriciated. I also wonder whether is there any such nonsolvable groups ?

By saying "symmetric", I mean lattice is isomorphic to its reverse lattice.

Example: Elemantary abelian $p$ groups, the groups that all Sylow subgroups of prime orders are such examples.

Note I had asked this question there.

  • $\begingroup$ @YCor: Thank you. I edited the question. $\endgroup$
    – mesel
    Jun 4 '15 at 20:12

You should have a look at Roland Schmidt's book Subgroup lattices of groups, especially Chapter 8, "Dualities of subgroup lattices". In this chapter, groups $G$ are studied such that there is another group $\overline{G}$ (a "dual"), such that the subgroup lattice of $G$ is isomorphic to the reverse of the subgroup lattice of of $\overline{G}$. A group with the property in your question is "self-dual" in Schmidt's terminology.
For example, a locally finite group with dual is metabelian and lattice-isomorphic to an abelian group and actually self-dual (Corollary 8.2.5), and so any finite group with dual is supersolvable, and every non-normal Sylow has prime order (Corollary 8.2.6). Thus a non-solvable example must be infinite. Indeed, Tarski monsters are non-solvable examples.


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