Group Algebra FG Research Articles

Group algebras and Lie nilpotence

Let ⁎ be an involution of a group algebra FG induced by an involution of the group G. For charF≠2, we classify the groups G with no 2-elements and with no nonabelian dihedral groups involved whose Lie algebra of ⁎-skew elements is nilpotent.

Lie derived length and involutions in group algebras

Let G be a group such that the set of p-elements of G forms a finite nonabelian subgroup, where p is an odd prime, and let F be a field of characteristic p. In this paper we prove that the lower bound of the Lie derived length of the group algebra FG given by Shalev in [11] is also a lower bound for the Lie derived length of the set of symmetric elements of FG for every involution which is linear extension of an involutive anti-automorphism of G. Furthermore, we provide counterexamples to the interesting cases which are not covered by the main theorem.

Nilpotency class of symmetric units of group algebras

Let F be a field of odd prime characteristic p, G a group, U the group of units in the group algebra FG, and U+ the subgroup of U generated by the elements of U fixed by the anti-automorphism of FG which inverts all elements of G. It is known that U is nilpotent 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 and only G' is cyclic and not a Sylow subgroup of G. Adalbert Bovdi and Janos 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 FP, 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 x Q formed with this P also leads to a U+ attaining the general bound, so here a necessary and sufficient condition remains elusive.

Group algebras of torsion groups and Lie nilpotence

Let ∗ be an involution of a group algebra FG induced by an involution of the group G. For char F ≠ 2, we classify the torsion groups G with no elements of order 2 whose Lie algebra of ∗-skew elements is nilpotent.

Separable algebras over infinite fields are 2-generated and finitely presented

We prove that every separable algebra over an infinite field F admits a presentation with 2 generators and finitely many relations. In particular, this is true for finite direct sums of matrix algebras over F and for group algebras FG, where G is a finite group such that the order of G is invertible in F. We illustrate the usefulness of such presentations by using them to find a polynomial criterion to decide when 2 ordered pairs of 2 × 2 matrices (A, B) and (A′, B′) with entries in a commutative ring R are automorphically conjugate over the matrix algebra M 2(R), under an additional assumption that both pairs generate M 2(R) as an R-algebra.

Group identities on symmetric units

Let F be an infinite field of characteristic different from 2, G a group and ∗ an involution of G extended by linearity to an involution of the group algebra FG. Here we completely characterize the torsion groups G for which the ∗-symmetric units of FG satisfy a group identity. When ∗ is the classical involution induced from g → g − 1 , g ∈ G , this result was obtained in [A. Giambruno, S.K. Sehgal, A. Valenti, Symmetric units and group identities, Manuscripta Math. 96 (1998) 443–461].

Warfield invariants in Abelian group algebras

LetF be a field of characteristicp≠0 andG an Abelian group. For each primeq and each ordinal α we calculate in terms ofG and its sections the Warfieldq-invariantsW α,q (V (FG)) of the groupV(FG) of all normalized units in the commutative group algebraFG when eitherq =p, G t /G p is infinite and F is perfect, orG is pmixed. Surprisingly, these invariants do not depend onF. This supplies recent result of ours in [3].

ON THE DERIVED LENGTH OF THE GROUP OF UNITS OF A GROUP ALGEBRA

Let G be a group with cyclic derived subgroup of p-power order (p > 2), and F a field of characteristic p. We determine the derived length of the group of units of the group algebra FG.

Group Algebras: Normal Subgroups and Ideals

The standard correspondence between the normal subgroups of the group G and some ideals of the group algebra FG is described. There is the problem of what we can say (or even prove) about a two-sided ideal \(I \neq (0)\) of \({\mathbb{F}}G\) that does not contain any element of the form 1 − g ≠ 0, g ∈G of the standard basis of the augmentation ideal of \({\mathbb{F}}G\) . The main part of the argument of [2] yields the insight that, for such an ideal I there exists an expansion \(H \supseteq G\) such that the ideal J of \({\mathbb{F}}H\) spanned by I contains an element 1 − h, h ∈ H \ G. Using the ideas of [2], we construct \({\mathbb{F}}\) -thick groups H such that for every ideal J ≠ (0) of \({\mathbb{F}}H\) there are elements 1 − h ≠ 0 in J. This construction allows many variations. Examples of simple \({\mathbb{F}}\) -thick groups were pointed out in [2]. A natural class of (in general non-simple) \({\mathbb{F}}\) -full groups are the normal sections of the groups $$S^{\prime} := \rm{Sym}({\it M})/ \rm{Fin}({\it M})\,\rm{for\,any\,infinite\,set} {\it M}.$$ (Here, Fin(M) is the subgroup of all finitary permutations of M.)

Commutative Group Algebras of Summable p-Groups

Let G be an Abelian group whose p-component G p is summable and let F be a field of characteristic p. Let S(FG) be the normalized unit p-group of the group algebra FG. The main results of the present article are that G p is a direct factor of S(FG) with totally projective complement, provided G p is of countable length and F is perfect; and, in particular, under these circumstances S(FG) is summable if and only if G p is summable. Moreover, if FG≅ FH as F-algebras for some group H and if G p is summable, then H p is summablale. These achievements improve results due to Hill and Ullery (1997) and also their generalizations given by Danchev (2000ab20012004).

A note on blocks with abelian defect groups

A recent result by H. Meyer shows that, for a field F of characteristic p > 0 and a finite group G with an abelian Sylow p-subgroup, the F-subspace Z p ′ F G of the group algebra FG spanned by all p-regular class sums in G is multiplicatively closed, i.e. a subalgebra of the center Z FG of FG. Here we generalize this result to blocks. More precisely, we show that, for a block A of a group algebra FG with an abelian defect group, the F-subspace Z p ′ A : = A ∩ Z p ′ F G is multiplicatively closed, i.e. a subalgebra of the center Z A of A. We also show that this subalgebra is invariant under perfect isometries and hence under derived equivalences.

Units of commutative group algebra with involution

Let p be an odd prime, F the field of p elements and G a finite abelian p-group with an arbitrary involutory automorphism. Extend this auto- morphism to the group algebra FG and consider the unitary and the symmetric normalized units of FG. This paper provides bases and determines the invariants of the two subgroups formed by these units.

Group algebras with an Engel group of units

AbstractLet F be a field of characteristic p and G a group containing at least one element of order p. It is proved that the group of units of the group algebra FG is a bounded Engel group if and only if FG is a bounded Engel algebra, and that this is the case if and only if G is nilpotent and has a normal subgroup H such that both the factor group G/H and the commutator subgroup H′ are finite p–groups.

Group algebras with symmetric units satisfying a group identity

We study group algebras FG for which the symmetric units under the natural involution: g *=g −1 satisfy a group identity. For infinite fields F of characteristic ≠2, a classification of torsion groups G whose symmetric units U +(FG) satisfy a group identity was given in [3] by Giambruno-Sehgal-Valenti. We extend this work to non torsion groups.

Group Algebras with a Solvable Group of Units#

A description is given of the structure modular group algebras FG of characteristic p whose group of units U(FG) is a solvable group.

Isomorphic Commutative Group Algebras of p-Mixed Warfield Groups

Let G be a p-mixed Warfield Abelian group and F a field of char F = p ≠ 0. It is proved that if for any group H the group algebras FH and FG are F-isomorphic, then H is isomorphic to G. This presentation enlarges a result of W. May argued when G is p-local Warfield Abelian and published in Proc. Amer. Math. Soc. (1988).

Simple components and central units in group algebras

Let G be a finite group. Berman [Dokl. Akad. Nauk 106 (1956) 767] and Witt [J. Reine Angew. Math. 190 (1952) 231] evaluate, independently, the number of simple components of the group algebra FG when F is a field of characteristic 0. In this paper we extend this result to fields of arbitrary characteristic which does not divide the order of G. We also compute the rank of the group of the central units of ZG and obtain an alternative proof of a well-known result of Ritter and Sehgal.

Generalized Commutativity in Group Algebras

AbstractWe study group algebras FG which can be graded by a finite abelian group Γ such that FG is β-commutative for a skew-symmetric bicharacter β on Γ with values in F*.

Commutative Group Algebras of Cardinality $${\cal N}$$ 1

We let FG be the group algebra of an abelian group G over a field F with characteristic p. Also, we define G p and S(FG) as the groups of all p-primary normed elements in G and FG, respectively. We prove that if G p is Hausdorff and both F and G have cardinalities not exceeding \({\cal N}\)1, then S(FG)/G p is a direct sum of cyclics. Thus G p is a direct factor of S(FG), and in particular G is a direct factor of the group of all normalized units V(FG), provided that the torsion part of G is a p-group. This answers a question posed by us in Hokkaido Math. J. (2000). Moreover we establish that if G is p-splitting, then any F-isomorphism of the group algebras FG and FH implies that H is p-splitting. We also show that if G is of power \({\cal N}\)1 whose p-component G p is a direct sum of torsion-complete groups and F has power p, then the F-isomorphism of FG and FH for any group H yields an isomorphism between G p and H p . In particular, when G is of power \({\cal N}\)1 and is p-mixed of torsion-free rank 1 whose G p is torsion-complete, we have G ≂ H. If F is in power p and G, with cardinality \({\cal N}\)1, is a direct sum of p-local algebraically compact groups such that FG ≂ FH as F-algebras for some group H, then G ≂ H. These statements extend results due to Beers-Richman-Walker (1983), and also partially solve a well-known question raised by May in 1979.

Group Identities on Units of Group Algebras

Let U be the group of units of the group algebra FG of a group G over a field F. Suppose that either F is infinite or G has an element of infinite order. We characterize groups G so that U satisfies a group identity. Under the assumption that G modulo the torsion elements is nilpotent this gives a complete classification of such groups. For torsion groups this problem has already been settled in recent years.