Strongly 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 is a prime number, and W is a conjugacy functor for the prime p. We say that W is a strongly closed conjugacy functor if it satisfies the following equivalent conditions:

  • There exists a p-Sylow subgroup P of G such that W(P) is a strongly closed subgroup of P relative to G.
  • For every p-Sylow subgroup P of G, W(P) is a strongly closed subgroup of P relative to G.

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 W(P) is a normal subgroup of G. |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
weakly closed conjugacy functor W(P) is weakly closed in P relative to G. |FULL LIST, MORE INFO