# Conjugacy functor that controls strong 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, further, that is a conjugacy functor on . We say that **controls strong fusion** on if, for any -Sylow subgroup of , is a subset-conjugacy-determined subgroup inside . In other words, given two subsets and in that are conjugate by , there exists such that conjugation by has the same effect as conjugation by on every element of .

## 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 |