The Enhanced Hasse diagram of the subgroup structure of the symmetries of the cube.

 This one took some work. Let me explain it.

 

The numbers along the left side give the order of the subgroups. There is one subgroup of order 1, nine subgroups of order 2, four subgroups of order 3, four subgroups of order 4, four subgroups of order 6, three subgroups of order 8, and only one subgroup of order 12 and one of order 24. 

 

Circles indicate normal subgroups. There are four, and their orders are 1, 4, 12 and 24.

 

Squares indicate subgroups that are not normal, and the ovals are subnormal.

 

Of the nine subgroups of order 2, three are generated by even permutations and six by odd permutations. The even permutation subgroups each connect to two subgroups of order 4. Each one is matched directly to one of the subnormal subgroups above it, while all are subgroups of the normal subgroup.

The six odd permutations generate subgroups of one of the order 8 subgroups and two of the order 6 subgroups. This creates a lot of traffic in the middle of the diagram, so I assigned a color of the rainbow to each of the six order 2 subgroups and all the arrows leading up to the higher levels from a single order 2 subgroup are the same color. For example, the first odd permutation order 2 subgroup is connected with red arrows to the second order 8 subgroup and the to the first and third order 6 subgroups.

 

None of these six subgroups are normal or even subnormal.

 

My next task is explaining the symmetries of the dodecahedron, a group that is isomorphic to the symmetries of the icosahedron. It is a group of order 60 and will be represented as a subgroup of S12. The subgroup structure is much more complex than the group we have been working on, and I'm not sure I'm up to making an enhanced Hasse diagram of it.

 

Time will tell.

  

Enhanced Hasse Diagrams and Subnormal Subgroups.

 So far, we have talked about subgroups being either normal or not normal, but there is a refinement of not normal groups called subnormal. A subgroup H is subnormal if it is not normal in G, but in a string of nested subgroups ending in G, each one normal in the supergroup directly above it.

We just started talking about the quaternion group, an eight element non-abelian group where all the subgroups are normal. The other eight element non-abelian group we have discussed is D4, the symmetries of the square. The order of a subgroup has to divide the order of the group, so the only possible sizes of subgroups of a group of order 8 are 4, 2 and 1.  In D4, the 8 element subgroup is the group itself, and the only subgroup of order 1 is the identity. While normality can be difficult to prove in some cases, one of the easiest cases is that if a group has order 2n, and subgroup of order n is normal. This means all the subgroups of order 4 must be normal, but we have to check on each of the subgroups of order 2. It turns out some are not normal, but if we look at them as 2 element subgroups of a group of order 4, they must be normal because 2 is half of 4.

 

Here is the Enhanced Hasse Diagram for D4. There are three subgroups of order 4, all of them normal, so they are represented by ovals. Of the five subgroups of order 2, only the one generated by R180° is normal, while the other four are subnormal. Subnormal groups are represented by rectangles with rounded corners.

 

Commentary

 

I am learning about subnormality on the fly as I put it up on the blog. D4 is a subgroup of S4, but it is not normal or subnormal, since subgroups of a symmetric group must be a union of conjugacy classes.  Written in cycle notation, D4 looks like this.

 

Rotations: (1), (1234), (13)(24), (1423)

Reflections: (12)(34), (14)(23), (13), (24)

 

Because the symmetric group has 24 elements and the dihedral group has 8, there is no subgroup "between" them, because there is no number between 8 and 24 that is divisible by 8 and also divides 24. D4 is not normal in S4 because it doesn't include all the 4-cycles and doesn't include all the 2-cycles. The chain of subgroups that defines subnormality is supposed to go all the way up to the group itself, and this is not the case here.

 

I expect there is a name for such a situation since it can be found so easily using well known small finite groups. I will search for it over the next few weeks, and I hope to report back.

 

In any case, learning new stuff is fun, but I fully expect I am reinventing the wheel here. It wouldn't be the first time.