On Symmetric Units in Group Algebras

Bovdi, Victor

doi:10.1081/agb-100107935

Cited by 11 publications

(5 citation statements)

References 5 publications

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“…We recall that in Bovdi (2001), a ring R is called G-favorable if for any g ∈ G, of finite order g , there is a nonzero ∈ R such that 1 − g is invertible in R. Notice that every infinite field is G-favorable. Proof.…”

Section: Symmetric Unitsmentioning

confidence: 99%

Symmetric Elements Under Oriented Involutions in Group Rings

Cristo

Milies

2006

Communications in Algebra

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Section: Symmetric Unitsmentioning

confidence: 99%

Symmetric Elements Under Oriented Involutions in Group Rings

Cristo

Milies

2006

Communications in Algebra

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“…They gave conditions under which the symmetric units form a subgroup in the case of locally finite -groups and a commutative ring. Bovdi and Parmenter [2] answered the same question for integral group rings in the case of periodic groups, and Bovdi [3] for a non-torsion nilpotent group and semiprime or a torsion group and a -favourable integral domain. In this last case he also proved that the symmetric units being a subgroup is equivalent to the symmetric elements being a subring.…”

Section: Introductionmentioning

confidence: 99%

Symmetric Elements of Nonlinear Involutions in Group Rings

García-Delgado

Raposo

2012

Algebra Colloq.

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Given an involution φ : G → G in a group G and a ring R, we study the extensions, not necessarily linear, to an involution ψ : RG → RG in the group ring RG. We investigate the symmetric elements, those α ∈ RG for which ψ(α) = α, and give necessary and sufficient conditions for the set of symmetric elements, (RG)ψ, to be a subring of RG. This work is a generalization of [6] and references therein where only linear extensions of the group involution are considered.

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“…We will denote by S + the set of symmetric elements of S ⊆ F G; that is, S + = {x ∈ S : x * = x}. A number of interesting results on the symmetric units of group rings can be found, for example, in the articles [4], [6], [7], [14], [15] and in the book [13]. This paper is devoted to the study of the nilpotency class of U + (F G).…”

Section: Introductionmentioning

confidence: 99%

“…We should remark that the assumption P to be powerful is not necessary for the equality. Using the LAGUNA [5] software package in the GAP [21] computer algebra system, it is easy to verify that if P is the noncommutative group of order 27 with exponent 3 and char F = 3, then the equality holds, although this group is not powerful.…”

Section: Introductionmentioning

confidence: 99%

Nilpotency class of symmetric units of group algebras

Balogh¹,

Juhász²

2011

Publ. Math. Debrecen

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Let F be a field of odd prime characteristic p, G a group, U the group of units in the group algebra F G, and U + the subgroup of U generated by the elements of U fixed by the anti-automorphism of F G which inverts all elements of G. It is known that U is nilpotent if G is nilpotent and the commutator subgroup G has p-power order, and then the nilpotency class of U is at most the order of G ; this bound is attained if and only if G is cyclic and not a Sylow subgroup of G. Adalbert Bovdi and János Kurdics proved the 'if' part of this last statement by exhibiting a nontrivial commutator of the relevant weight. For the special case when G is a nonabelian torsion group (so G cannot possibly be a Sylow subgroup), the present paper identifies such a commutator in U + , showing (Theorem 1) that the same bound is attained even by 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 nilpotent even though U is not. The torsion groups G which allow this are known (from the work of Gregory T. Lee) to be precisely the direct products of a finite p-group P , a quaternion group Q of order 8, and an elementary abelian 2-group. Theorem 2: in this case, the nilpotency class of U + is strictly smaller than the nilpotency index of the augmentation ideal of the group algebra F P , and this bound is attained whenever P is a powerful p-group. The nonabelian group P of order 27 and exponent 3 is not powerful, yet the G = P × Q formed with this P also leads to a U + attaining the general bound, so here a necessary and sufficient condition remains elusive.

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On Symmetric Units in Group Algebras

Cited by 11 publications

References 5 publications

Symmetric Elements Under Oriented Involutions in Group Rings

Symmetric Elements Under Oriented Involutions in Group Rings

Symmetric Elements of Nonlinear Involutions in Group Rings

Nilpotency class of symmetric units of group algebras

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