Characteristic p-functor that gives a characteristic subgroup

This article defines a property that can be evaluated for a characteristic p-functor in the context of a finite group.|View other 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 $p$ arising from a characteristic p-functor. We say that $W$ is a characteristic p-functor that gives a characteristic 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 characteristic subgroup of $G$.
   • For every $p$-Sylow subgroup $P$ of $G$, $W(P)$ is a characteristic subgroup of $G$.
5. 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