Conjugacy functor that controls strong fusion

From Groupprops
Jump to: navigation, search
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 and p is a prime number. Suppose, further, that W is a conjugacy functor on G. We say that W controls strong fusion on G if, for any p-Sylow subgroup P of G, P is a subset-conjugacy-determined subgroup inside N_G(W(P)). In other words, given two subsets A and B in P that are conjugate by g \in G, there exists h \in N_G(W(P)) such that conjugation by h has the same effect as conjugation by g on every element of A.

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 |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
conjugacy functor that controls fusion |FULL LIST, MORE INFO