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

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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).

Facts used

  1. Normality satisfies inverse image condition
  2. Frattini's argument

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]