Weakly closed conjugacy functor

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


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:
  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:

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).