Post #10: Two groups of order 8, the abelian group Z_8 and the non-abelian group D_4.

 The abelian group Z8 is as uncomplicated as a group of order 8 can get. It has a subgroup order 4, another of order 2, and a third of order 1, which is to say the identity. Let's represent it as an additive group and make the Hasse diagram of the subgroup structure.

The structure of subgroups is like nesting Russian dolls, where the next smallest subgroup always fits inside the previous set in the structure.

Easy peasy, lemon squeezy.

The Hasse diagram for D4 is much more complex. There are three subgroups of order 4 and five subgroups of order 8. The subgroup {I, R180°} is contained in every one of the order 4 subgroups. The only element all the order 2 subgroups have in common is the identity I.

 

We have not yet explored the group structure of the subgroups of order 4 that contain two reflections. Tomorrow, we will look at the Cayley table and Hasse diagram of this abelian group known as the Klein four-group or Z2×Z2.