Conjugacy functor that gives a normal subgroup

This article defines a property that can be evaluated for a conjugacy functor on a finite group. |View all such properties

Definition

Suppose $p$ is a prime number and $G$ is a finite group such that $W$ is a conjugacy functor for $G$ for the prime $p$. We say that $W$ is a conjugacy functor that gives a normal subgroup if it satisfies the following equivalent conditions:

1. For every pair of $p$-Sylow subgroups $P,Q$ of $G$, $W(P)=W(Q)$.
2. For every pair of $p$-Sylow subgroups $P,Q$ of $G$, $W(P)$ is a normal subgroup of $Q$.
3. Each of these:
   • $W$ is a weakly closed conjugacy functor and there exists a $p$-Sylow subgroup $P$ of $G$ such that $W(P)\leq O_{p}(G)$ where $O_{p}(G)$ is the p-core of $G$.
   • $W$ is a weakly closed conjugacy functor and for every $p$-Sylow subgroup $P$ of $G$, $W(P)\leq O_{p}(G)$ where $O_{p}(G)$ is the $p$-core of $G$.
4. Each of these:
   • There exists a $p$-Sylow subgroup $P$ of $G$ such that $W(P)$ is a normal subgroup of $G$.
   • For every $p$-Sylow subgroup $P$ of $G$, $W(P)$ is a normal subgroup of $G$.

Equivalence of definitions

Further information: equivalence of normality and characteristicity conditions for conjugacy functor

Relation with other properties

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
conjugacy functor that controls strong fusion	element-wise conjugacy in a $p$-Sylow subgroup $P$ is determined in the normalizer of $W(P)$			|FULL LIST, MORE INFO
conjugacy functor that controls fusion	subset-wise conjugacy in a $p$-Sylow subgroup $P$ is determined in the normalizer of $W(P)$			|FULL LIST, MORE INFO
conjugacy functor whose normalizer generates whole group with p'-core	For $P$ a $p$-Sylow subgroup, $O_{p'}(G)N_{G}(W(P))=G$			|FULL LIST, MORE INFO
strongly closed conjugacy functor	returns a subgroup that is a strongly closed subgroup in the Sylow subgroup relative to the whole group.			|FULL LIST, MORE INFO
weakly closed conjugacy functor	returns a subgroup that is a weakly closed subgroup in the Sylow subgroup relative to the whole group.			|FULL LIST, MORE INFO