Weakly closed conjugacy functor

Definition
Suppose $$G$$ is a finite group, $$p$$ a prime number, and $$W$$ a defining ingredient::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:
 * 2) * There exists a $$p$$-Sylow subgroup $$P$$ of $$G$$ such that $$W(P)$$ is a defining ingredient::weakly closed subgroup of $$P$$ relative to $$G$$.
 * 3) * For every $$p$$-Sylow subgroup $$P$$ of $$G$$, $$W(P)$$ is a weakly closed subgroup of $$P$$ relative to $$G$$.
 * 4) Either of these equivalent:
 * 5) * There exists a $$p$$-Sylow subgroup $$P$$ such that, for every $$p$$-Sylow subgroup $$Q$$ containing $$W(P)$$, $$W(P) = W(Q)$$.
 * 6) * For every $$p$$-Sylow subgroup $$P$$, and for every $$p$$-Sylow subgroup $$Q$$ containing $$W(P)$$, $$W(P) = W(Q)$$.
 * 7) Either of these equivalent:
 * 8) * 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 $$Q$$.
 * 9) * 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 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 $$N_G(W(P)) = N_G(P)$$.