# Equivalence of definitions of characteristic p-functor whose normalizer generates whole group with p'-core

From Groupprops

This article gives a proof/explanation of the equivalence of multiple definitions for the term characteristic p-functor whose normalizer generates whole group with p'-core

View a complete list of pages giving proofs of equivalence of definitions

## Contents

## Statement

Suppose is a group, is a prime number, and is a characteristic p-functor. The following are equivalent:

- For one (and hence every) -Sylow subgroup of ,
- For one (and hence every) -Sylow subgroup of , the image of in the quotient is a normal subgroup of .
- For one (and hence every) -Sylow subgroup of , is a normal subgroup of .
- For one (and hence every) -Sylow subgroup of , is a characteristic subgroup of .

## Facts used

- Equivalence of definitions of conjugacy functor whose normalizer generates whole group with p'-core
- Equivalence of normality and characteristicity conditions for conjugacy functor

## Proof

### Equivalence of (1) and (2)

This is direct from Fact (1). Note that that *only* uses the conjugacy functor.

### Equivalence of (2) and (3)

This requires the observation that for the quotient map , Sylow subgroups of map isomorphically to Sylow subgroups of , so the quotient map commutes with .

### Equivalence of (3) and (4)

This is direct from Fact (2).