Maximal among abelian characteristic subgroups may be multiple and isomorphic
The example described here also shows many other things:
Example involving the upper triangular matrices
Suppose is any prime, and let be the group of upper-triangular unipotent matrices over the field of elements. Let be the subgroup of comprising those matrices where the entry is zero. Then, is a group of order .
By fact (1), we have that has two subgroups that are Abelian of maximum order: the rectangle groups of dimensions and respectively. Call these subgroups and respectively. Then, observe that:
- Both and are also Abelian subgroups of maximum order in . Moreover, they are the only Abelian subgroups of maximum order in since they are the only Abelian subgroups of maximum order in .
- and are isomorphic -- in fact, they are conjugate subgroups inside the bigger group . This conjugation restricts to an automorphism of , but not of .
- Both and are normal in , and hence in . The quotient is isomorphic to the unipotent subgroup of 3-by-3 matrices while the quotient is isomorphic to the elementary Abelian group of order . Hence, and are not automorphic subgroups in .
- Thus, and are the only Abelian subgroups of their order, and they are not automorphic in . Hence, they are both characteristic subgroups. Since they are Abelian of maximum order, they are both maximal among Abelian characteristic subgroups.
We have thus established all the required conditions.