# Conjugacy functor that controls fusion

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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

## Contents

## Definition

Suppose is a finite group and is a prime number. Suppose is a conjugacy functor on the nontrivial -subgroups of . We say that controls -fusion in if, for any -Sylow subgroup of , is a weak subset-conjugacy-determined subgroup inside .

(Note that is contained in because is normal in by the conjugation-invariance property that conjugacy functors have to satisfy. In fact, by the fact that conjugacy functor gives normalizer-relatively normal subgroup).

## 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 | Conjugacy functor that controls strong fusion, Conjugacy functor whose normalizer generates whole group with p'-core|FULL LIST, MORE INFO | |||

Conjugacy functor that controls strong fusion | |FULL LIST, MORE INFO | |||

Conjugacy functor whose normalizer generates whole group with p'-core | Conjugacy functor whose normalizer generates whole group with p'-core controls fusion | |FULL LIST, MORE INFO |

### Related group properties

## Facts

- Control of fusion is local: If is a conjugacy functor such that the restriction of to the normalizer of any non-identity subgroup controls fusion in that subgroup, then controls fusion in the whole group.