The modular group algebra problem for metacyclic 𝑝-groups

A deformation-theoretic version of Maschke's theorem for modular group algebras: The commutative case

On monomial codes in modular group algebras

In 1978, Robert McEliece constructed the first asymmetric code-based cryptosystem using noise-immune Goppa codes; no effective key attacks has been described for it yet. By now, quite a lot of code-based cryptosystems are known; however, their cryptographic security is inferior to that of the classical McEliece cryptosystem. In connection with the development of quantum computing, code-based cryptosystems are considered as an alternative to number theoretical ones; therefore, the problem of seeking promising classes of codes to construct new secure code-based cryptosystems is relevant. For this purpose, noncommutative codes can be used, that is, ideals in group algebras $${{\mathbb{F}}_{q}}G$$ over finite noncommutative groups $$G$$. The security of cryptosystems based on codes induced by subgroup codes has been studied earlier. The Artin–Wedderburn theorem, which proves the existence of an isomorphism of a group algebra to the direct sum of matrix algebras, is important for studying noncommutative codes. However, the particular form of terms and the construction of the isomorphism are not specified by this theorem; thus, for each group, there remains the problem of constructing the Wedderburn representation. The complete Wedderburn decomposition for the group algebra $${{\mathbb{F}}_{q}}{{D}_{{2n}}}$$ over the dihedral group $${{D}_{{2n}}}$$ has been obtained by F.E. Brochero Martinez in the case when the cardinality of the field and the order of the group are relatively prime numbers. Using these results, we study codes in the group algebra $${{\mathbb{F}}_{q}}{{D}_{{2n}}}$$ in this paper. The problem on the structure of all codes is solved, and the structure of codes induced by codes over cyclic subgroups of $${{D}_{{2n}}}$$ is described, which is of interest for cryptographic applications.

Modular group algebras with centrally metabelian unit groups

In this paper, we classify the modular group algebra [Formula: see text] of a group [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text] having upper Lie nilpotency index [Formula: see text] for [Formula: see text] and [Formula: see text]. Group algebras of upper Lie nilpotency index [Formula: see text] for [Formula: see text], have already been characterized completely.

In this paper, we classify the modular group algebra [Formula: see text] of a group [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text] having upper Lie nilpotency index [Formula: see text]. The group algebra [Formula: see text] with [Formula: see text] has already been described.

Let p be a prime, R a commutative ring of characteristic p , G a p -primary abelian group, and V(RG) the group of normalized units in the group algebra RG . It is largely determined when V(RG) is totally projective. This generalizes the well-known special case when R is a perfect field. Moreover, the current proof has the advantage of greater clarity.

We prove certain results pertaining to some isomorphism properties of commutative modular group algebras and briefly review a paper by pointing out some obvious mistakes and essential incorrectness.

We study the unitary subgroupV *(F2 G) in the group algebras F2 Gof 2-groups of maximal class over the field F2of two elements. We show that there does not exist a normal complement to Gin V *(F2 G) if and only if Gis the dihedral group of order 2n(n≥ 5) or the semidihedral group of order 2n(n≥5). We also describe V *(F2 G) when the subgroup Ghas order 8 and 16 and in this case there exist a normal complement of Gin V *(F2 G).

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.

We study the Modular Isomorphism Problem applying a combination of existing and new techniques. We make use of the small group algebra to give a positive answer for two classes of groups of nilpotency class 3. We also introduce a new approach to derive properties of the lower central series of a finite 𝑝-group from the structure of the associated modular group algebra. Finally, we study the class of so-called 𝑝-obelisks which are highlighted by recent computer-aided investigations of the problem.

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.

Let 𝐹 be a field of odd characteristic and 𝐺 a group for which the group algebra F ⁢ G FG is modular. In this paper, necessary and sufficient conditions are presented for the commutator subgroup of the unit group of F ⁢ G FG to be nilpotent of class at most 2.

Let X X be a smooth curve over a finite field of characteristic p p , let E E be a number field, and let L = { L λ } \mathbf {L} = \{\mathcal {L}_\lambda \} be an E E -compatible system of lisse sheaves on the curve X X . For each place λ \lambda of E E not lying over p p , the λ \lambda -component of the system L \mathbf {L} is a lisse E λ E_\lambda -sheaf L λ \mathcal {L}_\lambda on X X , whose associated arithmetic monodromy group is an algebraic group over the local field E λ E_\lambda . We use Serre’s theory of Frobenius tori and Lafforgue’s proof of Deligne’s conjecture to show that when the E E -compatible system L \mathbf {L} is semisimple and pure of some integer weight, the isomorphism type of the identity component of these monodromy groups is “independent of λ \lambda ”. More precisely, after replacing E E by a finite extension, there exists a connected split reductive algebraic group G 0 G_0 over the number field E E such that for every place λ \lambda of E E not lying over p p , the identity component of the arithmetic monodromy group of L λ \mathcal {L}_\lambda is isomorphic to the group G 0 G_0 with coefficients extended to the local field E λ E_\lambda .

For modular abelian group algebras over a perfect field of prime characteristic, a connection is given between the direct factor problem and the isomorphism problem.