P-solvable not implies Glauberman type for p

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This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., p-solvable group) need not satisfy the second group property (i.e., group of Glauberman type for a prime)
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It is possible to have a prime number p and a finite group G such that G is p-solvable (i.e., is a p-solvable group) but not a group of Glauberman type for p.

In fact, this is possible for the primes p = 2,3.

Related facts


Example at the prime two

Further information: symmetric group:S4

Consider the prime p = 2 and the group G = S_4, the symmetric group on the set \{ 1,2,3,4 \}. G is a solvable group, and hence is 2-solvable. However, G is not of Glauberman type for the prime two, because if we take P as the 2-Sylow subgroup generated by \{ (1,2,3,4), (1,3) \}, then Z(J(P)) = \{ (), (1,3)(2,4)\}. In this case, we have:

O_{2'}(G)N_G(Z(J(P))) = P \ne G.

Example at the prime three