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

This article gives a proof/explanation of the equivalence of multiple definitions for the term conjugacy 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 conjugacy functor for $G$. The following conditions are equivalent, where $P$ is any $p$-Sylow subgroup of $G$.

1. $O_{p'}(G)N_G(W(P)) = G$
2. The image of $W(P)$ in the quotient $G/O_{p'}(G)$ is a normal subgroup of $G/O_{p'}(G)$.

## Proof

### (2) implies (1)

Given: A prime $p$, a finite group $G$ such that if $K = G/O_{p'}(G)$, then for every $p$-Sylow subgroup $P$ of $G$, the image of $W(P)$ is normal in $K$.

To prove: For one (and hence every) $p$-Sylow subgroup $P$ of $G$, $\! G = O_{p'}(G)N_G(W(P))$.

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 Let $H = O_{p'}(G)$, so $K = G/H$, and $\varphi:G \to G/H$ be the quotient map, with $\varphi(G) = G/H = K$. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
2 Let $L = \varphi^{-1}(\varphi(W(P))))$. In other words, $\! L = O_{p'}(G)W(P)$.
3 $L$ is normal in $G$. Fact (1) $\varphi(W(P))$ is normal in $K$. Steps (5), (6) fact-step combination direct.
4 $\! W(P)$ is $p$-Sylow in $\! L$. Step (2) [SHOW MORE]
5 $\! LN_G(W(P)) = G$. Fact (2) Steps (3), (4) Fact-step-combination direct.
6 $\! O_{p'}(G)N_G(W(P))) = G$ Steps (2), (5) [SHOW MORE]