# The groups with symmetric subgroups lattice

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.

• Does "symmetric" mean one of these things?
– anon
Jun 3, 2015 at 19:41
• @anon: Raughly speaking, Hasse Diagram of the lattice is same from Top to botton or botton to top. Jun 3, 2015 at 19:44
• @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. Jun 3, 2015 at 19:47
• @anon: I hope it is clear now ? Jun 3, 2015 at 19:51
• The lattices that you call 'symmetric' are usually called 'self-dual'. All finite abelian groups have self-dual subgroup lattices.
– Eran
Jun 23, 2015 at 14:16

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 ($$G = \hat{G}$$)...