Narrow Results

Let $FG$ be the group algebra of a finite $p$-group $G$ over a finite field $F$ of positive characteristic $p$. Let ⊛ be an involution of the algebra $FG$ which is a linear extension of an anti-automorphism of the group $G$ to $FG$. If $p$ is an odd prime, then the order of the ⊛-unitary subgroup of $FG$ is established. For the case $p=2,$ we generalize a result obtained for finite abelian $2$-groups. It is proved that the order of the ∗-unitary subgroup of $FG$ of a non-abelian $2$-group is always divisible by a number which depends only on the size of $F$, the order of $G$ and the number of elements of order two in $G$. Moreover, we show that the order of the ∗-unitary subgroup of $FG$ determines the order of the finite $p$-group $G$.

A classification of Lie solvable group algebras of derived length 3 over fields of characteristic 3 and 5 is given. It is known that for a field K with Char K ≥ 7 and G an arbitrary group, δ^[3](L(KG)) = 0 if and only if δ⁽³⁾(KG) = 0. Same turns out to be true when Char K = 5 and G is an arbitrary group or when Char K = 3 and G is a two-Engel group.

In this paper we provide a characterization of Lie solvable group algebras of derived length three over a field of characteristic three when G is a non-2-Engel group with abelian commutator subgroup.

We give a description of the primitive central idempotents of the rational group algebra ℚG of a finite group G. Such a description is already investigated by Jespers, Olteanu and del Río, but some unknown scalars are involved. Our description also gives answers to their questions.

Let K be a field of characteristic p > 0 and let G be an arbitrary group. In this paper, we classify group algebras KG which are strongly Lie nilpotent of index at most 8. We also show that for k ≤ 6, KG is strongly Lie nilpotent of index k if and only if it is Lie nilpotent of index k.

Given a partial action α of a group G on the group algebra FH, where H is a finite group and F is a field whose characteristic p divides the order of H, we investigate the associativity question of the partial crossed product FH _*α G. If FH _*α G is associative for any G and any α, then FH is called strongly associative. We characterize the strongly associative modular group algebras FH with H being a p-solvable group.

The conjecture on units of group algebras of a torsion-free supersoluble group is saying that every unit is trivial, i.e. a product of a nonzero element of the field and an element of the group. This conjecture is still open and even in the slightly simple case of the fours group $\mathrm{\Gamma }=〈x,y|{(x^2)}^y=x^{-2},{(y^2)}^x=y^{-2}〉$, it is not yet known. The main result of this paper is to show that a wide range of elements of group algebra of $\mathrm{\Gamma }$ are nonunit.

Several relations and bounds for the dimension of principal ideals in group algebras are determined by analyzing minimal polynomials of regular representations. These results are used in the two last sections. First, in the context of semisimple group algebras, to compute, for any abelian code, an element with Hamming weight equal to its dimension. Finally, to get bounds on the minimum distance of certain MDS group codes. A relation between a class of group codes and MDS codes is presented. Examples illustrating the main results are provided.

In this paper, we address the question of finding the length of group algebras of finite abelian groups in the case when the characteristic of the ground field divides the order of the group. We evaluate the exact length for an arbitrary abelian $p$-group. For other groups we provide upper and lower bounds for the length of their group algebras.

In this paper we solve the problem of finding the length of group algebras of arbitrary finite abelian groups in the case when the characteristic of the ground field does not divide the order of the group. We show that these group algebras have maximal possible lengths for infinite fields and sufficiently large finite fields since they are one-generated. In case of small fields we prove that the length is bounded from above by a logarithmic function of the order of the group.

Let ${𝒜}$ be a semiprime Banach algebra and let $\phi$ be a nonzero character on ${𝒜}$. In this paper, we discuss the projectivity of certain left Banach ${𝒜}$-modules in terms of left $\phi$-contractibility of Banach algebras. We also apply our results for the category of quantum group algebras and eventually, we present illustrative examples of the projective left Banach modules over the group algebra $L^1(G)$ and the Fourier algebra $A(G)$ of a locally compact group $G$.

Let KG be the group algebra of a group G over a field K of positive characteristic p, and let 𝔇_(n)(G) and 𝔇_[n](G) denote the n-th upper Lie dimension subgroup and the n-th lower one, respectively. In [1] and [12], the equality 𝔇_(n)(G) =𝔇_[n](G) is verified when p ≥ 5. Motivated by [16, Problem 55], in the present paper we establish it for particular classes of groups when p ≤ 3. Finally, we introduce and study a new central series of G linked with the Lie nilpotency class of KG.

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that if KG is Lie nilpotent, then its upper (or lower) Lie nilpotency index is at most |G′| + 1, where |G′| is the order of the commutator subgroup. The class of groups G for which these indices are maximal or almost maximal has already been determined. Here we determine G for which upper (or lower) Lie nilpotency index is the next highest possible.

Let K be a normal subgroup of the finite group H. To a point of a K-interior H-algebra we associate a group extension, and we prove that this extension is isomorphic to an extension associated to the point given by the Brauer homomorphism. This may be regarded as a generalization and an alternative treatment of Dade's results [2, Section 12].

In this paper minimal codes for several classes of non-cyclic abelian groups have been constructed by explicitly determining a complete set of primitive idempotents in the corresponding group algebras. Some classes of non-p-groups have also been considered. The minimum distances of such abelian codes have been discussed and compared to the minimum distances of cyclic codes of same lengths and dimensions over the same field.

Let $G$ be a group and let $K$ be a field of characteristic $p>0$. In this paper, we have classified the group algebras $KG$ which are strongly Lie nilpotent of index $9$, $10$ or $11$.

For a group algebra $KG$ of a group $G$ over a field $K$ of characteristic $p>0$, it is well known that $p+1$ is the minimal upper as well as the minimal lower Lie nilpotency index. Group algebras of upper Lie nilpotency index upto $7p-5$ have already been characterized completely. In this paper, we classify the modular group algebra $KG$ having upper Lie nilpotency index $8p-6$ which is the possible next higher Lie nilpotency index.

In this paper, we classify the modular group algebra $KG$ of a group $G$ over a field $K$ of characteristic $p>0$ having upper Lie nilpotency index $t^L(KG)=9p-7$. The group algebra $KG$ with $t^L(KG)<9p-7$ has already been described.

Let $G$ be a group and let $K$ be a field of characteristic $p>0$. Lie nilpotent group algebras of strong Lie nilpotency index at most 13 have been classified by many authors. In this paper, our aim is to classify the group algebras $KG$ which are strongly Lie nilpotent of index 14.

In this paper, we establish the structure of the unit group of the group algebra $FG$ where $G$ is an abelian group of order at most $16$ and $F$ is a finite field of characteristic $p$ with $q=p^n$ elements.