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On the Nilpotency Index of the Radical of a Group Algebra, III
Author(s) -
Motose Kaoru
Publication year - 1982
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-25.1.39
Subject(s) - index (typography) , citation , mathematics , algebra over a field , group (periodic table) , library science , computer science , chemistry , pure mathematics , world wide web , organic chemistry
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: