Self-normalizing Sylow subgroup

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: self-normalizing subgroup and Sylow subgroup
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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 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.

Relation with other properties

Weaker properties

• Abnormal subgroup: Also related:
  • Weakly abnormal subgroup
  • Self-normalizing subgroup
• Abnormal Hall subgroup: Also related:
  • Self-normalizing Hall subgroup
  • Pronormal Hall subgroup
• Sylow subgroup
• Carter subgroup