Well-placed subgroup

Definition

Suppose $G$ is a finite group, $p$ is a prime number, and $P$ is a $p$-Sylow subgroup of $G$. Suppose $W$ is a conjugacy functor on $G$ with respect to the prime $p$.

A subgroup $H$ of $P$ is termed well-placed in $P$ relative to $G$ if the following holds.

Consider the following sequence:

$W_{1}(H)=H,P_{1}(H)=N_{P}(H),N_{1}(H)=N_{G}(H)$,

and the recursive definition is:

$W_{i+1}(H)=W(P_{i}(H)),P_{i+1}(H)=N_{P}(W_{i+1}(H)),N_{i+1}(H)=N_{G}(W_{i+1}(H))$.

Then, we require that each $P_{i}(H)$ is a $p$-Sylow subgroup of $N_{i}(H)$. In other words, we require that each $W_{i}(H)$ is a subgroup whose normalizer in the Sylow is Sylow in the normalizer.

For a subgroup that is not well-placed, its height is defined as the smallest $d$ for which $P_{d}(H)$ is not $p$-Sylow in $N_{d}(H)$.

Relation with other properties

Weaker properties

• Subgroup whose normalizer in the Sylow is Sylow in the normalizer

Facts

• Every p-subgroup is well-placed in some Sylow subgroup
• Normal subgroup of Sylow subgroup is well-placed