Equivalence of definitions of group of Glauberman type for a prime

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This article gives a proof/explanation of the equivalence of multiple definitions for the term group of Glauberman type for a prime
View a complete list of pages giving proofs of equivalence of definitions

Statement

Suppose G is a finite group and p is a prime number. In the discussion below, Z(J(P)) denotes the subgroup obtained by applying the ZJ-functor to P. The ZJ-functor is defined as the center of the Thompson subgroup J(P), which in turn is defined as the join of abelian subgroups of maximum order. The following are equivalent:

  1. For one (and hence every) p-Sylow subgroup P of G, G = O_{p'}(G)N_G(Z(J(P))): Here, O_{p'}(G) denotes the p'-core of G,
  2. For one (and hence every) p-Sylow subgroup P of G, the image of Z(J(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), Z(J(Q)) is a normal subgroup of K.
  4. For one (and hence every) p-Sylow subgroup Q of K = G/O_{p'}(G), Z(J(Q)) is a characteristic subgroup of K.

Facts used

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

Proof

The proof follows directly from Fact (1), where the characteristic p-functor that we use is the ZJ-functor.