The Enhanced Hasse Diagrams for D_4 and the Quaterion group Q_8

The two non-abelian groups of order 8, D4 and Q8, do not need Simplified Enhanced Hasse diagrams. There are so few subgroups, simplification seems unnecessary. So instead, we get Enhanced Hasse Diagrams, which show the differences in their structures.

 

D4 has a total of ten subgroups, six of them are normal and the other four subnormal. There are two different copies of the Klein-4 group, {(1), (12)(34), (13)(24), (14)(23)} and {(1), (13), (24), (13)(24)}. They are isomorphic, but the second version has two odd permutations and two even, while the first version is all even permutations.

 

All subgroups, other than the Klein-4 subgroups and D4 itself,  are defined by a single generator.

The Q8 diagram much less cluttered, with only six subgroups total and all of them normal. In abelian groups, all subgroups are normal, but it is rare when this is true in a non-abelian group. Only Q8 itself is not define by a single generator.

 

An enhancement for Hasse diagrams of groups

 We have already talked about Hasse diagrams. The term for this kind of drawing is a directed graph, which is to say dots connected by lines with arrows. The technical terms are nodes and directed edges.

 

In graph theory, nodes can be different colors or shapes, as they are in this sample flowchart. In this example, only Step 5 is in the diamond shape, which represents a decision point. Notice that it has a Yes arrow and a No arrow, so there are two different nodes that might come after Step 5, here called Step 6A and Step 6B. Every node has a way in and a way out, except for Start, which only has a exit and is known as a source, and Stop, which does have a way out and is called a sink.

 

In the Hasse diagram of a group, the whole group is the source and the identity element is the sink. Since this is a diagram about subgroup structure, it would be nice if we could tell at a glance whether a subgroup was normal or not. My modest idea for an enhancement is to make the nodes ovals for the normal subgroups and rectangles for the non-normal subgroups.

In this Hasse diagram, I also introduce the shorthand method of writing a cyclic group. <(123)> means the subgroup generated by the permutation (123). (123)(123) = (132) and
(123)(123)(123) = (1). Putting the generator in pointy brackets is more concise than writing {(123), (132), (1)}.

Corresponding with Dan Jurca, he brought up a third type of subgroup that would be identified with yet another shape. We could have a nested chain of subgroups G ⊃ H ⊃ K, where K is normal in H, but not in G. I've been searching online and I can't find the terminology for an example like this. So far, I know it isn't called locally normal or semi-normal or partially normal. We have seen an example of a group that has subgroups with this property, and I hope by the weekend I will have found the correct term for this relatively common situation.

Commentary

If G ⊃ H, we can say H is a subgroup of G, and we can also say G is a supergroup of H. As someone who grew up listening to rock music, when I hear the word supergroup, I think of bands like Cream, The Traveling Wilburys or Crosby, Stills, Nash and Young.