P-solvable implies p-constrained

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This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., p-solvable group) must also satisfy the second group property (i.e., p-constrained group)
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Statement

Verbal statement

Any p-solvable group is a p-constrained group.

Statement with symbols

Suppose G is a finite group and p is a prime number. Suppose further that G is p-solvable. Then, if P is a p-Sylow subgroup, we have:

C_G(P \cap O_{p',p}(G)) \le O_{p',p}(G).

In other words, G is p-constrained.

Related facts

Converse

Facts used

  1. Equivalence of definitions of Sylow subgroup of normal subgroup: This states that the intersection of a Sylow subgroup and a normal subgroup is a Sylow subgroup of the normal subgroup.
  2. Sylow satisfies image condition
  3. Pi-separable and pi'-core-free implies pi-core is self-centralizing

Proof

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Given: A finite group G that is p-solvable for some prime p. P is a p-Sylow subgroup.

To prove: Let Q = P \cap O_{p',p}(G). Then, C_G(Q) \le O_{p',p}(G), where C_G(Q) is the centralizer of Q in G.

Proof: Let \varphi:G \to G/O_{p'}(G) be the natural quotient map. Note that \varphi^{-1}(O_p(G/O_{p'}(G))) = O_{p',p}(G).

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 G/O_{p'}(G) is p'-core-free [SHOW MORE]
2 Q is a p-Sylow subgroup of O_{p',p}(G) Fact (1) P is p-Sylow in G, and Q = P \cap O_{p',p}(G). [SHOW MORE]
3 \varphi(Q) = O_p(G/O_{p'}(G)), or equivalently \varphi^{-1}(\varphi(Q)) = O_{p',p}(G). Fact (2) Step (2) [SHOW MORE]
4 \varphi(Q) is self-centralizing, i.e., C_G(\varphi(Q)) \le \varphi(Q) = O_p(G/O_{p'}(G)) Fact (3) G is p-solvable. Steps (1), (3) [SHOW MORE]
5 \varphi(C_G(Q)) \le C_G(\varphi(Q)) This follows from the definition of homomorphism: if an element centralizes Q, its image centralizes the image of Q.
6 \varphi(C_G(Q)) \le \varphi(Q) Steps (4), (5) Step-combination direct
7 C_G(Q) \le \varphi^{-1}(\varphi(Q)) = O_{p',p}(G) Steps (3), (6) Step-combination direct