On the Nilpotency Index of the Radical of a Group Algebra, III

Let p be a fixed prime number, let K be a field of characteristic p, let G be a finite p-solvable group with a p-Sylow subgroup P of order pm, and let t(G) be the nilpotency index of the radical J(KG) of a group algebra KG of G over K. Moreover we set M= O,.(G), H= O,.,,(G), pr = 1 H/MI, and F/M a Frattini subgroup of H/M. Then G/H is isomorphic to a subgroup of GL(H/F) where H/F is regarded as a vector space over GF(p) (see [2, Lemma 1.2.53). This assertion shall be used freely in this paper without references. Every subgroup of GL(2, 3) acts naturally on the elementary abelian group E of order 9. Let T and S be a semidirect product of E by GL(2, 3) and SL(2, 3) with respect to this action,. respectively. Y. Tsushima [13] proved the inequality p” 2 t(G). In the light of this inequality he [ 143 (see also [ 1 l] ) proved that t(G) = pm if and only if P is cyclic. Further S. Koshitani [S, 61 (see also [ 111) has proved the following conditions are equivalent for p 2 3 and m 2 2: