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

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

## Statement

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

1. For one (and hence every) $p$-Sylow subgroup $P$ of $G$, $O_{p'}(G)N_G(W(P)) = G$
2. For one (and hence every) $p$-Sylow subgroup $P$ of $G$, the image of $W(P)$ in the quotient $G/O_{p'}(G)$ is a normal subgroup of $G/O_{p'}(G)$.
3. For one (and hence every) $p$-Sylow subgroup $Q$ of $K = G/O_{p'}(G)$, $W(Q)$ is a normal subgroup of $K$.
4. For one (and hence every) $p$-Sylow subgroup $Q$ of $K = G/O_{p'}(G)$, $W(Q)$ is a characteristic subgroup of $K$.

## 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 $G \to G/O_{p'}(G)$, Sylow subgroups of $G$ map isomorphically to Sylow subgroups of $G/O_{p'}(G)$, so the quotient map commutes with $W$.

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

This is direct from Fact (2).