| Co-Investigator(Kenkyū-buntansha) |
YASUMOTO Taichi Aichi Univ.of Edu.Fac.of Edu.AP, 教育学部, 助教授 (00231647)
WATANABE Osamu Aichi Univ.of Edu.Fac.of Edu.P, 教育学部, 教授 (30024011)
KANEMITSU Mitsuo Aichi Univ.of Edu.Fac.of Edu.P, 教育学部, 教授 (60024014)
KAWAMOTO Naoki Japan Coast Guard Acad.P, 教授
NINOMIYA Yasushi Shinshu Univ.Fac.of Sci.P, 理学部, 教授 (40092887)
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| Research Abstract |
Let t be a nonnegative integer, S be a finite 2-group and U be an elementary abelian normal subgroup of S. We write U ∈ μ_t(S) if U = <u ; |[u, A]| 【less than or equal】 2^t> for any elementary abelian subgroup A of S with [U, A, A] = 1. Let G be a finite group, S ∈ Syl_2(G) and V = <u^q ; u ∈ U, g ∈G>. Hayashi gave an estimation of m such that V ∈ μ_m(S) whenever V is abelian. Let k be a field of characteristic p>0 and G be a finite group. The nilpotency index t(G) of the Jacobson radical of the group algebra k[G] is one of the most important invariant to investigate the group algebra. Let G be a finite p-solvable group, P ∈ Syl_p (G) and p^m =|P|. Ninomiya classified all the groups satisfying p^<m-2> < t(G) < p^<m-1>. Moreover, when p = 3, he gave the example group with t(G) > t(P). This is a counterexample to the conjecture that t(G) 【less than or equal】 t(P) always holds. Kawamoto proved that there is a duality between the simplicity of commutative associative algebras and its derivation Lie algebras under some suitable conditions. Knemitsu gave a necessary and sufficient condition that an oversemigroup of a torsion-free cancellative semigroup S is flat over 5. Let G be a torsionfree additive group and S be a submonoid of G. He proved that If S is a GCD-semigroup which has prime ideals that are linearly ordered, then S is a valuation semigroup.
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