# 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 $W(P) \le O_p(G)$ 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 $G$, $W(P) \le O_p(G)$ 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 $W(P)$ Conjugacy functor that controls strong fusion, Conjugacy functor whose normalizer generates whole group with p'-core|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. Strongly closed conjugacy functor|FULL LIST, MORE INFO