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• W2029920161 abstract "Let F be a field of odd prime characteristic p, G a group, U the group of unitsin the group algebra FG, and U+ the subgroup of U generated by the elements of Ufixed by the anti-automorphism of FG which inverts all elements of G. It is knownthat U is nilpotent G is nilpotent and the commutator subgroup G' has p-powerorder, and then the nilpotency class of U is at most the order of G'; this bound is attained and only G' is cyclic and not a Sylow subgroup of G. Adalbert Bovdiand Janos Kurdics proved the if part of this last statement by exhibiting a nontrivialcommutator of the relevant weight. For the special case when G is a nonabelian torsiongroup (so G' cannot possibly be a Sylow subgroup), the present paper identifies sucha commutator in U+, showing (Theorem 1) that the same bound is attained evenby the nilpotency class of this subgroup. We do not know what happens when G'is not a Sylow subgroup but G is not torsion. It can happen that U+ is nilpotenteven though U is not. The torsion groups G which allow this are known (from thework of Gregory T. Lee) to be precisely the direct products of a finite p-group P, aquaternion group Q of order 8, and an elementary abelian 2-group. Theorem 2: in thiscase, the nilpotency class of U+ is strictly smaller than the nilpotency index of theaugmentation ideal of the group algebra FP, and this bound is attained wheneverP is a powerful p-group. The nonabelian group P of order 27 and exponent 3 is notpowerful, yet the G = P x Q formed with this P also leads to a U+ attaining thegeneral bound, so here a necessary and sufficient condition remains elusive." @default.
• W2029920161 created "2016-06-24" @default.
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• W2029920161 date "2011-07-01" @default.
• W2029920161 modified "2023-09-26" @default.
• W2029920161 title "Nilpotency class of symmetric units of group algebras" @default.
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• W2029920161 doi "https://doi.org/10.5486/pmd.2011.4999" @default.
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