Self-normalizing Sylow subgroup

Definition
A subgroup of a finite group is termed a self-normalizing Sylow subgroup if it satisfies the following equivalent conditions:


 * 1) It is a Sylow subgroup and is also self-normalizing: it equals its own normalizer in the whole group.
 * 2) It is a Sylow subgroup and is also weakly abnormal: every subgroup containing it is self-normalizing.
 * 3) It is a Sylow subgroup and is also abnormal.
 * 4) It is a Sylow subgroup that is also a defining ingredient::Carter subgroup: in other words, it is a nilpotent self-normalizing subgroup.
 * 5) It is a Carter subgroup that has prime power order.

The equivalence of definitions (1)-(3) follows from the fact that Sylow implies pronormal, and for pronormal subgroups, being self-normalizing, weakly abnormal, and abnormal are equivalent. The equivalence with (4) follows from the fact that prime power order implies nilpotent. The equivalence with (5) follows from the fact that in a finite group, there is exactly one conjugacy class of Carter subgroups (i.e., nilpotent self-normalizing subgroups) and that any $$p$$-subgroup that is not $$p$$-Sylow cannot be Carter because it is properly contained in its normalizer in any $$p$$-Sylow subgroup containing it.

Weaker properties

 * Stronger than::Abnormal subgroup: Also related:
 * Stronger than::Weakly abnormal subgroup
 * Stronger than::Self-normalizing subgroup
 * Stronger than::Abnormal Hall subgroup: Also related:
 * Stronger than::Self-normalizing Hall subgroup
 * Stronger than::Pronormal Hall subgroup
 * Stronger than::Sylow subgroup
 * Stronger than::Carter subgroup