9

Let $G$ be a finite group, and let $L(G)$ be the lattice of subgroups, partially ordered by inclusion. For example, below is $L(D_8)$.

$\quad\qquad\quad\qquad\quad\quad\qquad$enter image description here

I have two questions:

1) If we view the lattice as a directed graph, can non-isomorphic groups have isomorphic subgroup lattices?

2) Is there any software that computes subgroup lattices of finite groups? If so, in practice how large can the group be?

I'm interested in the subgroup lattice of the group $GL_n(\mathbb{F}_p)$.

Jared
  • 31,451

2 Answers2

8

1) Yes. A lattice isomorphism $f:\mathcal{L}(G) \to \mathcal{L}(H)$ is called a projectivity from $G$ to $H$. These are discussed in some detail in Roland Schmidt's book Subgroup lattices of groups. A projectivity is determined by its action on the cyclic subgroups (rather than on their generators, the elements of the groups).

A projectivity must map a cyclic group onto a cyclic group: being cyclic is a property of the lattice of subgroups. Similarly the class of finite groups and the class of locally finite groups are preserved by projectivities.

For instance, a group $G$ is a quasi-cyclic $p$-group (quasi-cyclic = cyclic or Prüfer $p$-group) if and only if the lattice of subgroups is a chain.

An isomorphism obviously induces a projectivity, but more subtly, certain bijective functions on groups that treat cyclic subgroups well (Theorem 1.3.1) induce projcetivities, and a projectivity is induced this way iff when $f(\langle g \rangle) = \langle h \rangle$, that $g$ and $h$ have the same order. As Ittay Weiss mentioned, this need not be the case, as cyclic groups of prime order show.

In some sense these are the “only” problem. Obviously the chain lattices (quasi-cyclic $p$-groups) don't depend on the prime, so each lattice is realized by infinitely many non-isomorphic groups. Lattices detect direct products of coprime order subgroups. Suzuki showed, that as long as a finite group does not have a cyclic direct factor, then only finitely many other groups (up to isomorphism) have the same subgroup lattice.

This is a survey of the first 40 pages of this 500+ page book, and I skipped some very nice results that don't directly answer your question.

2) Yes. GAP and Magma are the standard packages for working with large finite groups like $\operatorname{GL}(n,p)$. I find that GAP is easier to understand and modify, while Magma has faster implementations. For instance GAP takes 40 seconds to find the subgroup lattice of GL(4,2), while this takes under a second in Magma. Magma can find the subgroup lattice of GL(6,2), but I don't believe GL(7,2) is in practical range. Magma should not be greatly affected by changing the ground field, except for the explosion in the actual number of subgroups; however many of GAP's algorithms have exponential dependence on the ground field.

Make sure you really want the full subgroup lattice before asking for it. It is huge.

GL(4,2) has 48337 subgroups in 137 classes. GL(4,3) has 141224990 subgroups in 3085 classes. GL(5,2) has 54091780 subgroups in 1411 conjugacy classes. GL(5,3) has 10842232856355 subgroups in 93251 classes. See Enumerating all subgroups of the symmetric group for some asymptotics of the subgroup of permutation matrices.

Community
  • 1
Jack Schmidt
  • 55,589
  • Does GAP have some facilities for drawing subgroup lattices (perhaps the xgap package)? I remember a very long time ago, the walls of the Lehrstuhl D in Aachen (the birthplace of GAP) were all decorated with pictures of subgroup lattices. (But I don't think they got up to ${\rm GL}(4,2)$.) – Derek Holt Jul 21 '13 at 14:54
  • I use DotFileLatticeSubgroups (actually, I customize the code for the specific class of groups since lattices typically are ugly) to create a file that is better for printing. http://www.ms.uky.edu/~jack/2009-06-04-SmallGroupLattices/lattice.xhtml are the groups of order less than 64. I haven't kept up with XGAP, but I think it is being revitalized. – Jack Schmidt Jul 21 '13 at 14:59
  • @JackSchmidt: Thanks again Jack! Do either GAP or Magma report the isomorphism types of the subgroups? – Jared Jul 23 '13 at 02:54
  • @Jared: yes, IdGroup and StructureDescription in GAP. I use IdentifyGroup and ChiefFactors in Magma. See http://math.stackexchange.com/a/446105/583 for GAP. – Jack Schmidt Jul 23 '13 at 05:37
  • @JackSchmidt I have a question I think you may know: It is possible to write over the dot file and write with LaTeX over the dot file? If it is possible, how can I do that? – iam_agf Mar 02 '16 at 23:17
2

There are plenty of non-isomorphic groups with isomorphic lattices of subgroups. For instance, $L(\mathbb Z_p)$, for $p$ a prime, are all isomorphic.

The computer software Magma computes subgroup lattices. I don't know much about the algorithms that are used.

Ittay Weiss
  • 79,840
  • 7
  • 141
  • 236