Finite Group Algebras Research Articles

The Structure of the Unit Group of $\mathbb{F}_{3^{k}}(C_{3}\times D_{2n})$ and Finite Group Algebras of Groups of Order $42$

Let $\mathbb{F}_{q}$ be a finite field of characteristic $p>0$ with $|\mathbb{F}_{q}|=q=p^{k}$ and $\mathcal{U}(\mathbb{F}_{q}G)$ be the unit group of the group algebra $\mathbb{F}_{q}G$ for some group $G$. There are $6$ groups of order $42$ up to isomorphism. In this paper, we provide a characterization of $\mathcal{U}(\mathbb{F}_{3^{k}}(C_{3}\times D_{2n}))$ and establish the structures of the unit groups of some finite group algebras of groups of order $42$.

The isomorphism problem for rational group algebras of finite metacyclic nilpotent groups

. We prove that if G and H are finite metacyclic groups with isomorphic rational group algebras and one of them is nilpotent, then G and H are isomorphic.

Weak Hopf algebras, smash products and applications to adjoint-stable algebras

For a semisimple quasi-triangular Hopf algebra [Formula: see text] over a field [Formula: see text] of characteristic zero, and a strongly separable quantum commutative [Formula: see text]-module algebra [Formula: see text], we show that [Formula: see text] is a weak Hopf algebra, and it can be embedded into a weak Hopf algebra [Formula: see text]. With these structures, [Formula: see text] is the monoidal category introduced by Cohen and Westreich, and [Formula: see text] is tensor equivalent to [Formula: see text]. If [Formula: see text] is in the Müger center of [Formula: see text], then the embedding is a quasi-triangular weak Hopf algebra morphism. This explains the presence of a subgroup inclusion in the characterization of irreducible Yetter–Drinfeld modules for a finite group algebra.

Minimum depth of factorization algebra extensions

In this paper we study the minimum depth of a subalgebra embedded in a factorization algebra, and outline how subring depth, in this context, is related to module depth of the regular left module representation of the given subalgebra, within the appropriate module ring. As a consequence, we produce specific results for subring depth of a Hopf subalgebra in its Drinfel'd double. Moreover we study a necessary and sufficient condition for normality of a Hopf algebra within a double cross product context, which is equivalent to depth 2, as it is well known by a result of Kadison. Using the sufficient condition, we then prove some results regarding minimum depth 2 for Drinfel'd double Hopf subalgebra pairs, particularly in the case of finite group algebras. Finally, we provide formulas for the centralizer of a normal Hopf subalgebra in a double cross product scenario.

A short note on the Wedderburn components of a semisimple finite group algebra

One of the classical problems in the subject of group algebras is that of deducing Wedderburn decomposition of a finite semisimple group algebra. In this short note, we discuss how to check whether a matrix ring over a finite field is a Wedderburn component of the Wedderburn decomposition of a group algebra or not. Finally, we formulate an open problem in this direction.

Completeness of derivation algebras of finite commutative group algebras

Completeness of derivation algebras of finite commutative group algebras

Primitive decompositions of idempotents of the group algebras of dihedral groups and generalized quaternion groups

<p>In this paper, we introduce a method for computing the primitive decomposition of idempotents in any semisimple finite group algebra, utilizing its matrix representations and Wedderburn decomposition. Particularly, we use this method to calculate the examples of the dihedral group algebras $ \mathbb{C}[D_{2n}] $ and generalized quaternion group algebras $ \mathbb{C}[Q_{4m}] $. Inspired by the orthogonality relations of the character tables of these two families of groups, we obtain two sets of trigonometric identities. Furthermore, a group algebra isomorphism between $ \mathbb{C}[D_{8}] $ and $ \mathbb{C}[Q_{8}] $ is described, under which the two complete sets of primitive orthogonal idempotents of these group algebras correspond bijectively.</p>

Inverse images of block varieties

We extend a result due to Kawai on block varieties for blocks with abelian defect groups to blocks with arbitrary defect groups. Kawai’s result is a tool to calculate the cohomology variety of a module in a block B of a finite group algebra kG restricted to subgroups of a defect group P, provided that P is abelian. Kawai’s result coincides with a Theorem of Avrunin and Scott specialized to modules in the principal block and their restrictions to p-subgroups. J. Rickard raised the question whether Kawai’s result can be extended to modules in blocks with arbitrary defect groups. We show that this is indeed the case for modules whose corresponding module over some almost source algebra is fusion stable. We show that this fusion stability hypothesis is automatically satisfied for principal blocks and blocks with abelian defect groups.

On the normal complement problem of finite group algebra

Let [Formula: see text] be a group, [Formula: see text] be a finite field with a positive characteristic [Formula: see text], then [Formula: see text] is the group algebra of a group [Formula: see text] over [Formula: see text] with a positive characteristic [Formula: see text]. In this paper, the normal complement problem on semisimple group algebra [Formula: see text] is examined. The normal complement problem of groups of order up to 16 in their corresponding unit groups [Formula: see text] has already been discussed. We have proved that up to isomorphism all the three non-abelian groups of order 18 and up to isomorphism all three non-abelian groups of order 20 do not have normal complement in their corresponding unit groups [Formula: see text].

Crossed product codes

Bernal et al. provided a necessary and sufficient condition for a linear code to be realized as an ideal in a finite group algebra and De La Cruz and Willems proved a similar result for ideals in twisted group algebras. In this paper, we extend this characterization to crossed products. Furthermore, we determine conditions for some crossed product codes to be self-dual.

Evaluation of Some Algebraic Structures in Balanced Incomplete Block Design

Balanced incomplete block design is an incomplete block design in which any two varieties appear together an equal number of times. In algebra, the existence of block design is closely related to balanced incomplete block design. To ascertain the claim, this research aim to employ some algebraic structures to examine whether or not balanced incomplete block design is related to the above statement. The methods adopted are finite group, ring and field algebra. The result shows that balanced incomplete block design (BIBD) cannot form a finite group under multiplication binary operation, but it is for additive case. It is also revealed that balanced incomplete block design is not a field algebra in both binary operations no matter the size of the design, but it is a ring in all cases. In conclusion, BIBD of the form (X,B) is a semigroup, commutative group, semiring, commutative ring and subfield in both binary operations. Several theorems with proofs have been established in harmony with the algebraic structure mentioned above.

Left regular bands of groups and the Mantaci–Reutenauer algebra

We develop the idempotent theory for algebras over a class of semigroups called left regular bands of groups (LRBGs), which simultaneously generalize group algebras of finite groups and left regular band (LRB) algebras. Our techniques weave together the representation theory of finite groups and LRBs, opening the door for a systematic study of LRBGs in an analogous way to LRBs. We apply our results to construct complete systems of primitive orthogonal idempotents in the Mantaci–Reutenauer algebra MRn[G] associated to any finite group G. When G is abelian, we give closed form expressions for these idempotents, and when G is the cyclic group of order two, we prove that they recover idempotents introduced by Vazirani.

Group of Units of Finite Group Algebras for Groups of Order 24

Group of Units of Finite Group Algebras for Groups of Order 24

Existence of nonzero trace-zero idempotents in the group algebras of finite groups

Let [Formula: see text] be a finite group and [Formula: see text] a splitting field of [Formula: see text] of characteristic [Formula: see text]. Denote by [Formula: see text] the group algebra of [Formula: see text] over [Formula: see text] and [Formula: see text] the center of [Formula: see text]. Let [Formula: see text] be the [Formula: see text]-subspace of trace-zero elements of [Formula: see text]. We give some numerical sufficient and necessary conditions for [Formula: see text] and [Formula: see text], respectively, to be Mathieu subspaces of [Formula: see text] in terms of the degrees of irreducible representations of [Formula: see text] over [Formula: see text]. The same numerical conditions also characterize the finite groups [Formula: see text] that [Formula: see text] has no nonzero trace-zero idempotents and the finite groups [Formula: see text] that [Formula: see text] has no nonzero central trace-zero idempotents, respectively.

On isomorphisms between ideals of Fourier algebras of finite abelian groups

On isomorphisms between ideals of Fourier algebras of finite abelian groups

A short note on Wedderburn decomposition of a group algebra

In this paper, we extend the result of Mittal and Sharma (Bull. Korean Math. Soc. 2022) on Wedderburn decomposition (WD) of a finite semisimple group algebra. It is known that, under certain conditions, WD of a finite semisimple group algebra FqG can be computed from WD of its subalgebra Fq(G/H), where H is a normal subgroup of G of prime order and q=pk for some prime p and positive integer k. We extend this result to any normal subgroup H of G of order n.

Universal deformation rings of modules for generalized Brauer tree algebras of polynomial growth

Let k be an arbitrary field, be a k-algebra and V be a -module. When it exists, the universal deformation ring of V is a k-algebra whose local homomorphisms to R parametrize the lifts of V up to , where R is any complete, local commutative Noetherian k-algebra with residue field k. Symmetric special biserial algebras, which coincide with Brauer graph algebras, can be viewed as generalizing the blocks of finite type p-modular group algebras. Bleher and Wackwitz classified the universal deformation rings for all modules for symmetric special biserial algebras with finite representation type. In this paper, we begin to address the tame case. Specifically, let be any 1-domestic, symmetric special biserial algebra. By viewing as generalized Brauer tree algebras and making use of a derived equivalence, we classify the universal deformation rings for those -modules V with stable endomorphism ring isomorphic to k. The latter is a natural condition, since it guarantees the existence of the universal deformation ring .

Group of units of finite group algebras of groups of order 24

UDC 512.5 Let F be a finite field of characteristic p . The structures of the unit groups of group algebras over F of the three groups D 24 , S 4 and S L ( 2 , ℤ 3 ) of order 24 are completely described in numerous works. We present the unit groups of the group algebras over F for the remaining groups of order 24 , namely, C 24 , C 12 × C 2 , C 2 3 × C 3 , C 3 ⋊ C 8 , C 3 ⋊ Q 8 , D 6 × C 4 , C 6 ⋊ C 4 , C 3 ⋊ D 8 , C 3 × D 8 , C 3 × Q 8 , A 4 × C 2 , and D 12 × C 2 .

A Tour of p-Permutation Modules and Related Classes of Modules

This survey provides an overview of numerous results on p-permutation modules and the closely related classes of endo-trivial, endo-permutation and endo-p-permutation modules. These classes of modules play an important role in the representation theory of finite groups. For example, they are important building blocks used to understand and parametrise several kinds of categorical equivalences between blocks of finite group algebras. For this reason, there has been, since the late 1990’s, much interest in classifying such modules. The aim of this manuscript is to review classical results as well as all the major recent advances in the area. The first part of this survey serves as an introduction to the topic for non-experts in modular representation theory of finite groups, outlining proof ideas of the most important results at the foundations of the theory. Simultaneously, the connections between the aforementioned classes of modules are emphasised. In this respect, results, which are dispersed in the literature, are brought together, and emphasis is put on common properties and the role played by the p-permutation modules throughout the theory. Finally, in the last part of the manuscript, lifting results from positive characteristic to characteristic zero are collected and their proofs sketched.

Nontriviality of the first Hochschild cohomology of some block algebras of finite groups

We show that for some finite group block algebras, with nontrivial defect groups, the first Hochschild cohomology is nontrivial. Along the way we obtain methods to investigate the nontriviality of the first Hochschild cohomology of some twisted group algebras.