Examples of self-normalizing subgroups?

Question

I know that when $G$ is a $p$-group every proper subgroup is a proper subgroup of its normalizer. I was hoping to get some simple and not-so-simple examples of a subgroup that equals its normalizer. Thanks!

Answer 1

If a group is nilpotent, you won't be able to find such a subgroup unless you take the whole group. Otherwise, any non-normal maximal subgroup is self-normalising. (Note that a group is nilpotent if and only if every maximal subgroup is normal.)

Carter subgroups are another interesting source of examples of self-normalising subgroups. A subgroup $C$ is called a Carter subgroup if it is nilpotent and self-normalising. As such it is a maximal nilpotent subgroup, but not every maximal nilpotent subgroup is a Carter subgroup. Note that, in general, not every group has a Carter subgroup. For example, $A_5$ has no Carter subgroups.

The interesting theorem proved by Carter is this:

In a soluble group, there exist Carter subgroups and all Carter subgroups are conjugate.