On the generalized dimension subgroup problems

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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.