Maximal among abelian characteristic subgroups may be multiple and isomorphic
From Groupprops
Contents
Statement
It is possible to have a group of prime power order with two distinct subgroups
, such that both
and
are Maximal among abelian characteristic subgroups (?), and
.
Facts used
Related facts
The example described here also shows many other things:
Proof
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.