Conjugacy functor whose normalizer generates whole group with p'-core

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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 group, p is a prime number, and W is a conjugacy functor for G. We say that W is a conjugacy functor whose normalizer generates whole group with p'-core if it satisfies the following equivalent conditions for one (and hence every) p-Sylow subgroup P of G:

  1. O_{p'}(G)N_G(W(P)) = G
  2. The image of W(P) in the quotient G/O_{p'}(G) is a normal subgroup of G/O_{p'}(G).

Equivalence of definitions

Further information: equivalence of definitions of conjugacy functor whose normalizer generates whole group with p'-core

Related notions

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

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Conjugacy functor that controls fusion Conjugacy functor whose normalizer generates whole group with p'-core controls fusion