A note on transfer (Q1094533)

Let P be a finite p-group (p a prime). We define \(\Phi^*(P)=\cap_{M \max P}\Phi (M)\) where \(\Phi\) (M) is a Frattini subgroup of M. Since M/\(\Phi\) (M) is elementary abelian \(\exp (P/\Phi^*(P))\leq p^ 2\) and cl P/\(\Phi\) \({}^*(P)\leq p\); if cl P/\(\Phi\) \({}^*(P)=p\) then \(Z_ p wrZ_ p\) is a homomorphic image of \(P/\Phi^*(P)\). Th. 1. Let Q be a p- subgroup of the finite group G. Suppose that: (i) \(Q/\Phi^*(Q)\) has exponent p. (ii) The projective cover of the trivial module occurs as a summand of Q/\(\Phi\) (Q) viewed as a \(GF(p)N_ G(Q)/Q\)-module. Then \(Q\nleq G'\). Corr. 4. Let \(G=O^ 2(G)\), \(S\in Syl_ 2(G)\). Suppose that S has a maximal subgroup M such that \(M/\Phi^*(M)\) is elementary abelian. Then \(N_ G(S)/C_ G(S)\) and \(N_ G(M)/C_ G(M)\) cannot both be 2-groups.