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.