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In this paper, necessary and sufficient conditions on a group algebra of a finitely generated group G over a finite field K are determined such that the set of derivations of the group algebra form an associative K -algebra. The derivations of K G form a nontrivial associative K -algebra if and only if K has characteristic 2 and G is the direct product of a finite abelian group of odd order with either a cyclic 2-group or an infinite cyclic group. In this special case, the Jacobson radical of the resulting K -algebra is determined. [ABSTRACT FROM AUTHOR]

We endeavored to investigate directed strongly regular Cayley graphs (or DSRCG for short) over dicyclic groups Dic4n = ⟨α, β | αn = β4 = 1, β−1αβ = α −1 ⟩, where n is odd. We derived several DSRCGs over Dic4n for n odd. We then derived a criterion for a certain class of Cayley graph to be directed strongly regular. [ABSTRACT FROM AUTHOR]

Consider a finite field F q with characteristic p, where G is a crystallographic point group satisfying p ∤ | G | and q = p n . In this paper, we propose studying group codes in the crystallographic point group algebras F q G for the point groups C 2 h , C 6 v , and D 6 h . We compute the unique (linear and nonlinear) idempotents of F q G that correspond to the characters of the crystallographic point groups. These idempotents play a crucial role in characterizing the properties of the group codes. Based on the above results, we characterize the minimum distances and dimensions of the group codes. This provides valuable information about the error-correcting capabilities and the amount of information that can be transmitted through these codes. Furthermore, we construct MDS (Maximum Distance Separable) group codes and almost MDS group codes. [ABSTRACT FROM AUTHOR]

Let G and H be locally compact groups. We will show that each contractive Jordan isomorphism Φ : L 1 (G) → L 1 (H) is either an isometric isomorphism or an isometric anti-isomorphism. We will apply this result to study isometric two-sided zero product preservers on group algebras and, further, to study local and approximately local isometric automorphisms of group algebras. [ABSTRACT FROM AUTHOR]

Let F be an infinite field of characteristic p ≠ 2 , G be a group, and * be an involution of G extended linearly to an involution of the group algebra FG. In the literature, group identities on units U (FG) and on symmetric units U + (FG) = { α ∈ U (FG) | α * = α } have been considered. Here, we investigate normalized Laurent polynomial identities (as a generalization of group identities) on U + (FG) under the conditions that either p > 2 or F is algebraically closed. For instance, we show that if G is torsion and U + (FG) satisfies a normalized Laurent polynomial identity, then U + (FG) satisfies a group identity and FG satisfies a polynomial identity. [ABSTRACT FROM AUTHOR]

In this short note, the second dual of generalized group algebra (ℓ 1 (G , A) , ∗) equipped with both Arens product is investigated, where G is any discrete group and A is a Banach algebra containing a complemented algebraic copy of (ℓ 1 (N) , ∙) . We give an explicit family of annihilators(w.r.t both the Arens product) in the algebra ℓ 1 (G , A) ∗ ∗ , arising from non-principal ultrafilters on N and which are not in the toplogical center. As a consequence, we also deduce the fact that ℓ 1 (G , A) is not Strongly Arens irregular. [ABSTRACT FROM AUTHOR]

Knörr has constructed an ideal, in the center of the p-modular group algebra of a finite group G, whose dimension is the number of p-blocks of defect zero in G/Q; here p is a prime and Q is a normal p-subgroup of G. We generalize his construction to symmetric algebras. [ABSTRACT FROM AUTHOR]

A complete characterization of the unit group U(FqG) of the group algebra FqG of non-abelian group G of order 27 with exponent 3 over any finite field Fq is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we determine the structure of the unit groups of the semisimple group algebras of non-metabelian groups of order 144. Up to isomorphism, there are 197 non-isomorphic groups of order 144, and only 28 are non-metabelian. Mittal and Sharma [19] studied the unit groups of the semisimple group algebras of non-metabelian groups of order 144 that have exponent either 36 or 72. In this work, we characterize the unit groups of the group algebras of non-metabelian groups of order 144 having exponent 12 and 24. This paper completes the study of the unit groups of the group algebras of non-metabelian groups up to order 144. [ABSTRACT FROM AUTHOR]

In this paper, we consider the general linear group GL(2; 7) of 2×2 invertible matrices over the finite field of order 7 and compute the unit group of the semisimple group alge-bra FqGL(2; 7), where Fq is a finite field. For the computation of the unit group, we need the Wedderburn decomposition of FqGL(2; 7), which is determined by first computing the Wedder-burn decomposition of the group algebra Fq(PSL(3; 2) ... C2). Here PSL(3; 2) is the projective special linear group of degree 3 over a finite field of 2 elements. [ABSTRACT FROM AUTHOR]

In this paper, we focus on Brauer's height zero conjecture, Robinson's conjecture, and Olsson's conjecture regarding the direct product of finite groups and give relative versions of these conjectures by restricting them to the algebraic concept of the anchor group of an irreducible character. Consider G to be a finite simple group. We prove that the anchor group of the irreducible character of G with degree p is the trivial group, where p is an odd prime. Additionally, we introduce the relative version of the Green correspondence theorem with respect to this group. We then apply the relative versions of these conjectures to suitable examples of simple groups. Classical and standard theories on the direct product of finite groups, block theory, and character theory are used to achieve these results. [ABSTRACT FROM AUTHOR]

Let G be a finite p -group, and α an automorphism of the group algebra F p G. Then α fixes the socle of F p G pointwise. More generally, if k is a field of characteristic p , and α is a k -algebra automorphism of kG , then α induces a linear action on the dimension subquotients of the group, and the action on the socle is scalar multiplication by the (p − 1) st power of the product of the determinants of this action. The scalar is thus an element of (k ×) p − 1. [ABSTRACT FROM AUTHOR]

In this paper, we propose using quiver representations as a tool for understanding artificial neural network algorithms. Specifically, we construct these algorithms by utilizing the group algebra of a finite cyclic group as vertices and convolution transformations as maps. We will demonstrate the neural network using convolution operation in the group algebra. The convolution operation in the group algebra that is formed by a finite cyclic group can be seen as a circulant matrix. We will represent a circulant matrix as a map from a cycle permutation matrix to a polynomial function. Using the permutation matrix, we will see some properties of the circulant matrix. Furthermore, we will examine some properties of circulant matrices using representations of finite symmetric groups as permutation matrices. Using the properties, we also examine the properties of moduli spaces formed by the actions of the change of basis group on the set of quiver representations. Through this analysis, we can compute the dimension of the moduli spaces. [ABSTRACT FROM AUTHOR]

Consider an assignment of bits to the vertices of a connected graph Γ (V , E) with the property that the value of each vertex is a function of the values of its neighbors. A collection of such assignments is called a storage code of length |V| on Γ . In this paper we construct an infinite family of linear storage codes on triangle-free graphs with rates arbitrarily close to one. [ABSTRACT FROM AUTHOR]

In this paper, we develop a small cancellation theory for associative algebras with a basis of invertible elements. Namely, we study quotients of a group algebra of a free group and introduce three axioms for the corresponding defining relations. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a linear space and establish the corresponding structure theorems. We also provide a revision of a concept of Gröbner basis for our rings and establish a greedy algorithm for the Ideal Membership Problem. [ABSTRACT FROM AUTHOR]

We call an algebra A commutator-simple if [A,A] does not contain nonzero ideals of A. After providing several examples, we show that in these algebras derivations are determined by a condition that is applicable to the study of local derivations. This enables us to prove that every continuous local derivation D\colon L^1(G)\to L^1(G), where G is a unimodular locally compact group, is a derivation. We also give some remarks on homomorphism-like maps in commutator-simple algebras. [ABSTRACT FROM AUTHOR]

We consider G to be a finite group and p as a prime number. We fix ψ to be an irreducible character of G with its restriction to all p-regular elements of G and ψ 0 to be an irreducible Brauer character. The main aim of this paper is to describe and investigate the relationship between cyclic anchor group of ψ and the defect group of a p-block which contains ψ. Our methods are to study and generalize some facts for the cyclic defect groups of a p-block B to the case of a cyclic anchor group of irreducible characters which belong to B. We establish and prove a criteria for an irreducible character to have a cyclic anchor group. [ABSTRACT FROM AUTHOR]

Let $ \Bbb F_{q} $ be a finite field of characteristic $ q $ and $ S_n $ a symmetric group of order $ n! $. In this paper, group codes in the symmetric group algebras $ \Bbb F_{q}S_n $ with $ q > 3 $ and $ n = 3, 4 $ are proposed. We compute the unique (linear and nonlinear) idempotents of $ \Bbb F_q S_n $ corresponding to the characters of symmetric groups and use the results to characterize the minimum distances and dimensions of group codes. Furthermore, we construct MDS group codes and almost MDS group codes in $ \Bbb F_q S_3 $ and $ \Bbb F_q S_4 $.

In this paper, we characterize the structure of the unit group of the semisimple group algebras FqSL(2, 8) and FqSL(2, 9) of the special linear groups of 2 × 2 matrices with determinant 1 over the finite fields of order 8 and 9, respectively. [ABSTRACT FROM AUTHOR]

Let Cn, Qn and Dn be the cyclic group, the quaternion group and the dihedral group of order n, respectively. Recently, the structures of the unit groups of the finite group algebras of 2-groups that contain a normal cyclic subgroup of index 2 have been studied. The dihedral groups D2n, n ≥ 3 and the generalized quaternion groups Q4n, n ≥ 2 also contain a normal cyclic subgroup of index 2. The structures of the unit groups of the finite group algebras FQ12, FD12, F (C2 × Q12) and F(C2×D12) over a finite field F are well known. In this article, we continue this investigation and establish the structures of the unit groups of the group algebras F(Cn ×Q12) and F(Cn ×D12) over a finite field F of characteristic p containing pk elements. [ABSTRACT FROM AUTHOR]

Let FG be the semisimple group algebra of dihedral group G over the finite field F. In this paper, the complete set of primitive central idempotents of FG are expressed as elements of the group algebra. With the help of these primitive central idempotents, central codes are obtained. Further, with these central codes, non central codes are obtained. Also, the distances of these central and non central codes are compared. An illustration with the help of an example has also been discussed. [ABSTRACT FROM AUTHOR]

A ring A is (principally) nilary, denoted (pr-)nilary, if whenever XY = 0, then there exists a positive integer n such that either Xn = 0 or Yn = 0 for all (principal) ideals X, Y of A. We determine necessary and/or sufficient conditions for the group ring A[G] to be (principally) nilary in terms of conditions on the ring A or the group G. For example, we show that: (1) If A[G] is (pr-)nilary, then A is (pr-)nilary and either G is prime or the order of each finite nontrivial normal subgroup of G is nilpotent in A. (2) Assume that G is finite. Then G is nilpotent and A[G] is (pr-)nilary if and only if G is a p-group, char(A) = pα (p is a prime), and A is (pr-)nilary. (3) Let G be a finite supersolvable group such that q is the smallest prime dividing |G|, and F is a field with char(F) = q. Then F[G] is nilary if and only if G is a q-group. Examples are provided to illustrate and delimit our results. [ABSTRACT FROM AUTHOR]

Cohen and Taylor [On a certain Lie algebra defined by a finite group, Am. Math. Mon. 114 (2007) 633–639] introduced certain Lie algebras constructed using finite groups called the Plesken Lie algebras. In this paper, we describe the representation of these Plesken Lie algebras, their irreducibility and the Plesken Lie algebra modules. [ABSTRACT FROM AUTHOR]

Let N be a normal subgroup of a finite group G. From a result due to Brauer, it can be derived that the character table of G contains square submatrices which are induced by the G-conjugacy classes of elements in N and the G-orbits of irreducible characters of N. In the present paper, we provide an alternative approach to this fact through the structure of the group algebra. We also show that such matrices are non-singular and become a useful tool to obtain information of N from the character table of G. [ABSTRACT FROM AUTHOR]

In this article, the group algebra K[T]{\mathcal{K}}\left[{\mathscr{T}}] of the binary tetrahedral group T{\mathscr{T}} over a splitting field K{\mathcal{K}} of T{\mathscr{T}} with char(K)≠2,3{\rm{char}}\left({\mathcal{K}})\ne 2,3 is studied and the unique idempotents corresponding to all seven characters of the binary tetrahedral group are computed. Furthermore, the minimum weights and dimensions of various group codes generated by linear and nonlinear idempotents in this group algebra are characterized to establish these group codes.

Abstract We provide the structure of the unit group of $${\mathbb {F}}_{p^k}S_n$$ F p k S n , where $$p>n$$ p > n is a prime and $$S_n$$ S n denotes the symmetric group on n letters. We also provide the complete characterization of the unit group of the group algebra $${\mathbb {F}}_{p^k}A_6$$ F p k A 6 for $$p\ge 7$$ p ≥ 7 , where $$A_6$$ A 6 is the alternating group on 6 letters.

Certain classical codes can be viewed isomorphically as ideals of group algebras, while studying their algebraic structures help extracting the code properties. Research has shown that this was remarkably efficient in the case when the code generators are idempotents. In quantum error correction, the theory of stabilizer formalism requires classical self-orthogonal additive codes over the finite field GF(4), which, via the lens of group algebras, are essentially F 2 -submodules over GF(4). Therefore, this paper provides a classification on idempotents in commutative group algebra GF(4)G, followed by a criterion that allows idempotents to generate stabilizer subgroups. Later, the construction of quantum stabilizer codes is done in the case when G is a cyclic group C n , for n = 2 m - 1 and n = 2 m + 1 . Quantum bounds on their burst error minimum distance are subsequently determined. [ABSTRACT FROM AUTHOR]

In this paper, we focus on Brauer’s height zero conjecture, Robinson’s conjecture, and Olsson’s conjecture regarding the direct product of finite groups and give relative versions of these conjectures by restricting them to the algebraic concept of the anchor group of an irreducible character. Consider G to be a finite simple group. We prove that the anchor group of the irreducible character of G with degree p is the trivial group, where p is an odd prime. Additionally, we introduce the relative version of the Green correspondence theorem with respect to this group. We then apply the relative versions of these conjectures to suitable examples of simple groups. Classical and standard theories on the direct product of finite groups, block theory, and character theory are used to achieve these results.

In this paper, we propose using quiver representations as a tool for understanding artificial neural network algorithms. Specifically, we construct these algorithms by utilizing the group algebra of a finite cyclic group as vertices and convolution transformations as maps. We will demonstrate the neural network using convolution operation in the group algebra. The convolution operation in the group algebra that is formed by a finite cyclic group can be seen as a circulant matrix. We will represent a circulant matrix as a map from a cycle permutation matrix to a polynomial function. Using the permutation matrix, we will see some properties of the circulant matrix. Furthermore, we will examine some properties of circulant matrices using representations of finite symmetric groups as permutation matrices. Using the properties, we also examine the properties of moduli spaces formed by the actions of the change of basis group on the set of quiver representations. Through this analysis, we can compute the dimension of the moduli spaces.

We consider G to be a finite group and p as a prime number. We fix ψ to be an irreducible character of G with its restriction to all p-regular elements of G and ψ0 to be an irreducible Brauer character. The main aim of this paper is to describe and investigate the relationship between cyclic anchor group of ψ and the defect group of a p-block which contains ψ. Our methods are to study and generalize some facts for the cyclic defect groups of a p-block B to the case of a cyclic anchor group of irreducible characters which belong to B. We establish and prove a criteria for an irreducible character to have a cyclic anchor group.

In this paper, central and noncentral codes of the semisimple dihedral group algebra over finite fields are computed. [ABSTRACT FROM AUTHOR]

We characterize those modular group algebras FG whose group of unitary units is locally nilpotent or nilpotent under the classical involution of FG. The main results of [19, 20] are trivial consequences of this note. [ABSTRACT FROM AUTHOR]

We provide the structure of the unit group of F p k S n , where p > n is a prime and S n denotes the symmetric group on n letters. We also provide the complete characterization of the unit group of the group algebra F p k A 6 for p ≥ 7 , where A 6 is the alternating group on 6 letters. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some properties of ideals in group algebras of finite groups over fields. First, we highlight an important link between their dimension, their minimal Hamming distance and the group order. This is a generalized version of an uncertainty principle shown in 1992 by Meshulam. Secondly, we introduce the notion of the Schur product of ideals in group algebras and investigate the module structure and the dimension of the Schur square. We give a structural result on ideals that coincide with their Schur square, and we provide conditions for an ideal to be such that its Schur square has the projective cover of the trivial module as a direct summand. This has particularly interesting consequences for group algebras of 푝-groups over fields of characteristic 푝. [ABSTRACT FROM AUTHOR]

Let A be an algebra whose group of units U(A) satisfies a Laurent polynomial identity (LPI). We establish conditions on LPIs in such a way that nil-generated algebras and group algebras with torsion groups over infinite fields in characteristic p > 0 have nonmatrix identities. We also determine, in the context of group algebras with arbitrary LPI for the group of units, the existence of polynomial identities. [ABSTRACT FROM AUTHOR]

We classify, in terms of the structure of the finite group G, all group algebras KG for which all right ideals are right annihilators of principal left ideals. This means in the language of coding theory that we classify code-checkable group algebras KG which have been considered so far only for abelian groups G. Optimality of checkable codes and asymptotic results are discussed. [ABSTRACT FROM AUTHOR]

Let F be an infinite field and R be an algebraic F -algebra. It is known that if (R) , the unit group of R , satisfies a group identity, then R satisfies a polynomial identity. In this paper, using additive mappings, we generalize this result to some kinds of identities. Particularly, assume that ∗ is an involution of R. We show that if (R) satisfies a ∗-group identity, then R satisfies a polynomial identity. [ABSTRACT FROM AUTHOR]

A class of network codes have been proposed in the literature where the symbols transmitted on network edges are binary vectors and the coding operation performed in network nodes consists of the application of (possibly several) permutations on each incoming vector and XOR-ing the results to obtain the outgoing vector. These network codes, which we will refer to as permute-and-add network codes, involve simpler operations and are known to provide lower complexity solutions than scalar linear network codes. The complexity of these codes is determined by their degree which is the number of permutations applied on each incoming vector to compute an outgoing vector. Constructions of permute-and-add network codes for multicast networks are known. In this paper, we provide a new framework based on group algebras to design permute-and-add network codes for arbitrary (not necessarily multicast) networks. Our framework allows the use of any finite group of permutations (including circular shifts, proposed in prior works) and admits a trade-off between coding rate and the degree of the code. Further, our technique permits elegant recovery and generalizations of the key results on permute-and-add network codes known in the literature. [ABSTRACT FROM AUTHOR]

Let function P be a probability on a finite group G, i.e. $P(g)\geq0\ $ $(g\in G),\ \sum\limits_{g}P(g)=1$ (we write $\sum\limits_{g}$ instead of $\sum\limits_{g\in G})$. Convolution of two functions $P, \; Q$ on group $G$ is \linebreak $ (P*Q)(h)=\sum\limits_{g}P(g)Q(g^{-1}h)\ \ (h\in G)$. Let $E(g)=\frac{1}{|G|}\sum\limits_{g}g$ be the uniform (trivial) probability on the group $G$, $P^{(n)}=P*...*P$ ($n$ times) an $n$-fold convolution of $P$. Under well known mild condition probability $P^{(n)}$ converges to $E(g)$ at $n\rightarrow\infty$. A lot of papers are devoted to estimation the rate of this convergence for different norms. Any probability (and, in general, any function with values in the field $R$ of real numbers) on a group can be associated with an element of the group algebra of this group over the field $R$. It can be done as follows. Let $RG$ be a group algebra of a finite group $G$ over the field $R$. A probability $P(g)$ on the group $G$ corresponds to the element $ p = \sum\limits_{g} P(g)g $ of the algebra RG. We denote a function on the group $G$ with a capital letter and the corresponding element of $RG$ with the same (but small) letter, and call the latter a probability on $RG$. For instance, the uniform probability $E(g)$ corresponds to the element $e=\frac{1}{|G|}\sum\limits_{g}g\in RG. $ The convolution of two functions $P, Q$ on $G$ corresponds to product $pq$ of corresponding elements $p,q$ in the group algebra $RG$. For a natural number $n$, the $n$-fold convolution of the probability $P$ on $G$ corresponds to the element $p^n \in RG$. In the article we study the case when a linear combination of two probabilities in algebra $RG$ equals to the probability $e\in RG$. Such a linear combination must be convex. More exactly, we correspond to a probability $p \in RG$ another probability $p_1 \in RG$ in the following way. Two probabilities $p, p_1 \in RG$ are called complementary if their convex linear combination is $e$, i.e. $ \alpha p + (1- \alpha) p_1 = e$ for some number $\alpha$, $0

Let $ F $ be a finite field of characteristic $ p $ having $ q = p^n $ elements and $ G $ be an abelian group. In this paper, we determine the structure of the group of units of the group algebra $ FG $, where $ G $ is an abelian group of order $ 17\leq |G|\leq 20 $.

We prove that the Isomorphism Problem for group algebras reduces to group algebras over finite extensions of the prime field. In particular, the Modular Isomorphism Problem reduces to finite modular group algebras. [ABSTRACT FROM AUTHOR]

C ⁎ -algebras, group algebras, and the algebra A (X) of approximable operators on a Banach space X having the bounded approximation property are known to be zero product determined. In this paper we give a quantitative estimate of this property by showing that, for the Banach algebra A , there exists a constant α with the property that for every continuous bilinear functional φ : A × A → C there exists a continuous linear functional ξ on A such that sup ‖ a ‖ = ‖ b ‖ = 1 ⁡ | φ (a , b) − ξ (a b) | ≤ α sup ‖ a ‖ = ‖ b ‖ = 1 , a b = 0 ⁡ | φ (a , b) | in each of the following cases: (i) A is a C ⁎ -algebra, in which case α = 8 ; (ii) A = L 1 (G) for a locally compact group G , in which case α = 60 27 1 + sin ⁡ π 10 1 − 2 sin ⁡ π 10 ; (iii) A = A (X) for a Banach space X having property (A) (which is a rather strong approximation property for X), in which case α = 60 27 1 + sin ⁡ π 10 1 − 2 sin ⁡ π 10 C 2 , where C is a constant associated with the property (A) that we require for X. [ABSTRACT FROM AUTHOR]