Weakly closed conjugacy functor

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

Definition

Suppose $G$ is a finite group, $p$ a prime number, and $W$ a conjugacy functor on $G$ with respect to $p$. We say that $W$ is weakly closed in $G$ with respect to $p$ if the following equivalent conditions are satisfied:

1. Either of these equivalent:
   • There exists a $p$-Sylow subgroup $P$ of $G$ such that $W(P)$ is a weakly closed subgroup of $P$ relative to $G$.
   • For every $p$-Sylow subgroup $P$ of $G$, $W(P)$ is a weakly closed subgroup of $P$ relative to $G$.
2. Either of these equivalent:
   • There exists a $p$-Sylow subgroup $P$ such that, for every $p$-Sylow subgroup $Q$ containing $W(P)$, $W(P)=W(Q)$.
   • For every $p$-Sylow subgroup $P$, and for every $p$-Sylow subgroup $Q$ containing $W(P)$, $W(P)=W(Q)$.
3. Either of these equivalent:
   • There exists a $p$-Sylow subgroup $P$ of $G$ such that for any $p$-Sylow subgroup $Q$ of $G$ containing $W(P)$, $W(P)$ is a normal subgroup of $Q$.
   • For every $p$-Sylow subgroup $P$ of $G$, it is true that for any $p$-Sylow subgroup $Q$ of $G$ containing $W(P)$, $W(P)$ is a normal subgroup of $Q$.

For instance, a p-normal group is a group in which the conjugacy functor that arises by taking the center is weakly closed.

Equivalence of definitions

Further information: equivalence of definitions of weakly closed conjugacy functor

Equivalence relation induced on the set of Sylow subgroups

Given a weakly closed conjugacy functor $W$ for a prime $p$, we obtain an equivalence relation on the set $\operatorname {Syl} _{p}(G)$ of all $p$-Sylow subgroups of $G$. The equivalence relation is as follows: two $p$-Sylow subgroups $P,Q$ are equivalent if they satisfy the above equivalent conditions, for instance, $W(P)=W(Q)$ (this is the equivalent formulation that makes it easiest to see that the relation is reflexive, symmetric, and transitive).

This equivalence relation partitions the set $\operatorname {Syl} _{p}(G)$ into equivalence classes. It further turns out that all equivalence classes have the same size, because the conjugation with $G$ permutes them transitively. Moreover:

• The equivalence classes are parametrized by the conjugacy class of $W(P)$. The number of such equivalence classes is $[G:N_{G}(W(P))]$ and the size of each equivalence class is $[N_{G}(W(P)):N_{G}(P)]$.
• The equivalence class corresponding to a particular $W=W(P)$ is characterized as precisely those $p$-Sylow subgroups of $G$ that contain $W$.
• By the congruence condition on index of subgroup containing Sylow-normalizer, both the number of orbits and the size of each orbit are congruent to 1 modulo $p$.

Two extreme cases are of interest:

• The case that the equivalence relation has only one equivalence class, which means that $W(P)=W(Q)$ for all $P,Q\in \operatorname {Syl} _{p}(G)$, or equivalently, the subgroup $W(P)$ is inside $O_{p}(G)$, the p-core. This is equivalent to $W(P)$ being a normal subgroup of $G$. For more, see conjugacy functor that gives a normal subgroup.
• The case that the equivalence relation has equivalence classes all of size one, i.e., this is the case that $N_{G}(W(P))=N_{G}(P)$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
conjugacy functor that gives a normal subgroup				|FULL LIST, MORE INFO
strongly closed conjugacy functor				|FULL LIST, MORE INFO