On finite groups with a sylowp-subgroup of type (m,n)

A finitep-groupP is of type (m, n) ifP has nilpotency classm-1,P/P'≌Z _p ⁿ ×Z _p ⁿ and all the lower central factorsK _i (P)/K _i+1 (P) are cyclic of orderp ⁿ . Our main result on finite groups with a Sylowp-subgroup of type (m, n) is (Theorem 4.1): Let G be a finite group with a Sylow p-subgroup P of type (m, n), n≧2 p≧3, m≧(n+5)(p?1)+1. For H≦G denote \(\bar H = HO_{p'} (G)/O_{p'} (G)\) . If O_p (G) is not cyclic and P'₁ ≠ 1, then \(\bar P \Delta \bar G\) and \(\bar G = \bar P \cdot \bar T\) is a semidirect product of \(\bar P\) and \(\bar T\) , where \(\bar T\) is cyclic of order t, t|p-1. Here P₁ is the subgroup defined in section 0. This theorem easily yields that under its assumptionsN _G (P)/O ^P (N _G (P))≌G/O ^P (G), it gives information on the conjugacy pattern ofp-elements ofG and gives information on the structure ofp-local subgroups ofG (Theorems 4.2, 4.3 and 4.4).