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

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

Any reference and observation would be appriciated.

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

  • $\begingroup$ Does "symmetric" mean one of these things? $\endgroup$
    – anon
    Jun 3, 2015 at 19:41
  • $\begingroup$ @anon: Raughly speaking, Hasse Diagram of the lattice is same from Top to botton or botton to top. $\endgroup$
    – mesel
    Jun 3, 2015 at 19:44
  • $\begingroup$ @anon: Precisesly, If we put another order "$\leq *$" by $H\leq * K$ if $K\leq H$ then the new lattice is isomorphic to the previous one. $\endgroup$
    – mesel
    Jun 3, 2015 at 19:47
  • $\begingroup$ @anon: I hope it is clear now ? $\endgroup$
    – mesel
    Jun 3, 2015 at 19:51
  • 1
    $\begingroup$ The lattices that you call 'symmetric' are usually called 'self-dual'. All finite abelian groups have self-dual subgroup lattices. $\endgroup$
    – Eran
    Jun 23, 2015 at 14:16

1 Answer 1


This is covered in Chapter 8 of Schmidt's book "Subgroup lattices of groups"; everything in this answer is there, but I summarize it here for convenience.

Say a group $G$ has a dual $\hat{G}$ if their subgroup lattices are anti-isomorphic. Your question is about when $G$ is dual to itself.

Baer (1939) proved that every group w/ dual is torsion, i.e. all its elements have finite order.

For locally finite groups (which means every finitely generated subgroup is finite), there is a complete classification of groups with duals. (This is Theorem 8.2.2 in Schmidt, due to Suzuki and Zacher):

If $G$ is a locally finite group, and $\delta$ is a lattice anti-isomorphism to $L(\hat{G})$, then:

  1. $G = \prod_{\lambda} G_{\lambda}$ where the $G_{\lambda}$ are coprime to one another
  2. $\hat{G} = \prod_{\lambda} \delta(\hat{G}_\lambda)$, where $\hat{G}_\lambda := \langle G_{\mu} : \mu \neq \lambda \rangle$ and the $\delta(\hat{G}_\lambda)$ are coprime to one another
  3. For every $\lambda$, there exists (not nec. distinct) primes $p,q$ such that one of the following holds:
  • $G_\lambda$ is a cyclic $p$-group, $\delta(\hat{G}_\lambda)$ is a cyclic $q$-group, or
  • $G_\lambda$ and $\delta(\hat{G}_\lambda)$ are both either elementary abelian of order $p^n$ or a semi-direct product of an elementary abelian group of order $p^{n-1}$ by a group of order $q$, or
  • $G_\lambda$ and $\delta(\hat{G}_\lambda)$ are finite non-Hamiltonian $p$-groups with modular subgroup lattices

Maybe from these more can be said when $G$ is self-dual ($G = \hat{G}$)...


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .