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

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:
  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