Copyright (c) 2003-2004 James B. Wilson | Show Frames | Hide Frames

Planar Group Lattices

There are very few classes of groups whose subgroup lattices can be drawn as planar graphs. Here is the class of groups which are -- a proof that the list is complete is lacking, sorry.

The Outerplanar Groups

The following diagrams are the lattices of the only outerplanar groups.

A group is called outerplanar if its subgroup lattice can be drawn so that the resulting graph is outerplanar. This means all the subgroup vertices lie on the unbounded face of the graph embedded in the plane.

The Classification of Outerplanar Groups

C_{p^i} C_{p^i} x C_q C_{p^\infty} C_{p^\infty} x C_q

The Planar Groups

Planar Group Theory

The general classification of planar groups involves a particular case of the extension problem. For example, we look at the group C_p x C_p and attempt to "glue" the diamond shapes together along the exterior edges. Along the way an important restriction is determined: with the exception of a few exceptional cases, there are no acyclic groups with a unique minimal subgroup, or with a unique maximal subgroup. This means we do not need to consider groups that "stack" a planar group atop a chain of groups, or below. That is, the following lattices are impossible for groups.

Unique Maximal Subgroup Unique Minimal Subgroup Impossible Lattice
With the exception of the general quaternions, we can even reduce the lower chain to length one, so that outside of Q_{4n}, the following is impossible.
Unique Minimal Subgroup
It is also a short exercise to show there are no planar groups with three distinct prime divisors of the order. Thus all planar groups are of the form p^i q^j.

Planar p-groups

  i = 0, 1, 2 i = 3 i = 4 i > 4
p=2 All C8
C4 x C2
D8
Q8
C16
C8 x C2
C8 x_t C2
Q16
C2^i
C2^{i-1} x C2
C2^{i-1} x_t C2

p>2 All Cp^3
Cp^2 x Cp
Cp^2 x_t Cp
Cp^4
Cp^3 x Cp
Cp^3 x_t Cp
Cp^i
Cp^{i-1} x Cp
Cp^{i-1} x_t Cp
C_{p^{i-1}}\rtimes C_p

Planar p^i q-groups