In this article, we study the metacyclic p-group codes arising from finite semisimple group algebras. In [CM25], we studied group codes arising from metacyclic groups with order divisible by two distinct odd primes. In the current work, we focus on metacyclic p-group codes, as a result of which we are also able to extend the results of [CM25] for metacyclic groups with order divisible by any two primes, not necessarily odd or distinct. Consequently, existing results on group algebras of some important classes of groups, including dihedral and quaternion groups, have been extended. Additionally, we provide left codes for the undertaken group algebras. Finally, we construct non-central codes using units motivated by Bass and bicyclic units, which are inequivalent to any abelian group codes and yield best known parameters.
For a group $G$ and a subgroup $H$ of $G$ this article discusses the normalizer of $H$ in the units of a group ring $RG$. We prove that $H$ is only normalized by the `obvious' units, namely products of elements of $G$ normalizing $H$ and units of $RG$ centralizing $H$, provided $H$ is cyclic. Moreover we show that the normalizers of all subgroups of certain nilpotent and metacyclic groups in the corresponding group rings are as small as possible. These classes contain all dihedral groups, all finite nilpotent groups and all finite groups with all Sylow subgroups being cyclic.
Suppose that a Lie type algebra L over a field K admits a Frobenius group of automorphisms FH with cyclic kernel F of order n and complement H such that the fixed-point subalgebra of F is trivial and the fixed-point subalgebra of H is nilpotent of class c. If the ground field K contains a primitive n-th root of unity, then L is nilpotent and the nilpotency class of L is bounded in terms of |H| and c. The result extends the known theorem of Khukhro, Makarenko and Shumyatsky on Lie algebras with metacyclic Frobenius group of automorphisms.
We study the right and left commutation semigroups of finite metacyclic groups with trivial centre. These are presented \[G(m,n,k) = \left\langle {a,b;{a^m} = 1,{b^n} = 1,{a^b} = {a^k}} \right\rangle \quad (m,n,k\in\mathbb{Z}^+)\] with $(m,k - 1) = 1$ and $n = in{d_m}(k),$ the smallest positive integer for which ${k^n} = 1\,\pmod m,$ with the conjugate of $a$ by $b$ written ${a^b}( = {b^{ - 1}}ab).$ The \emph{right} and \emph{left commutation semigroups of} $G,$ denoted ${\rm P}(G)$ and $Λ(G),$ are the semigroups of mappings generated by $ρ(g):G \to G$ and $λ(g):G \to G$ defined by $(x)ρ(g) = …
For each prime number $s$ we introduce examples of $(s-1)$- and $s$-representation infinite algebras in the sense of Herschend, Iyama and Oppermann, which arise from skew group algebras of some metacyclic groups embedded in $\mathrm{SL}(s,\mathbb{C})$ and $\mathrm{SL}(s+1,\mathbb{C})$. For this purpose, we give a description of the McKay quiver with a superpotential of such groups. Moreover we show that for $s=2$ our examples correspond to the classical tame hereditary algebras of type $\tilde{D}$.
Let $K$ be a field of characteristic $p$ and $G$ a nonabelian metacyclic finite $p$-group. We give an explicit list of all metacyclic $p$-groups $G$, such that the group algebra $KG$ over a field of characteristic $p$ has a filtered multiplicative $K$-basis. We also present an example of a non-metacyclic 2-group $G$, such that the group algebra $KG$ over any field of characteristic 2 has a filtered multiplicative $K$-basis.
The object of study in this paper is the finite groups whose integral group rings have only trivial central units. Prime-power groups and metacyclic groups with this property are characterized. Metacyclic groups are classified according to the central height of the unit groups of their integral group rings.
Suppose that a Lie algebra $L$ admits a finite Frobenius group of automorphisms $FH$ with cyclic kernel $F$ and complement $H$ such that the characteristic of the ground field does not divide $|H|$. It is proved that if the subalgebra $C_L(F)$ of fixed points of the kernel has finite dimension $m$ and the subalgebra $C_L(H)$ of fixed points of the complement is nilpotent of class $c$, then $L$ has a nilpotent subalgebra of finite codimension bounded in terms of $m$, $c$, $|H|$, and $|F|$ whose nilpotency class is bounded in terms of only $|H|$ and $c$. Examples show that the condition of the k…
Finite non-abelian non-metacyclic $2$-generated $p$-groups (${p>2}$) of nilpotency class $2$ with cyclic commutator subgroup which are the additive groups of local nearrings are described. It is shown that the subgroup of all non-invertible elements is of index $p$ in its additive group.
For any number field K, it is unknown which finite groups appear as Galois groups of extensions L/K such that L is a maximal subfield of a division algebra with center K (a K-division algebra). For K=Q, the answer is described by the long standing Q-admissibility conjecture. We extend a theorem of Neukirch on embedding problems with local constraints in order to determine for every number field K, what finite solvable groups G appear as Galois groups of tame maximal subfields of K-division algebras, generalizing Liedahl's theorem for metacyclic G and Sonn's solution of the Q-admissibility conj…
We establish a complete classification of binary group codes with complementary duals for a finite group and explicitly determine the number of linear complementary dual (LCD) cyclic group codes by using cyclotomic cosets. The dimension and the minimum distance for LCD group codes are explored. Finally, we find a connection between LCD MDS group codes and maximal ideals.
We give a simple construction of codes from left ideals in group algebras of certain dihedral groups and give an example to show that they can produce codes with weights equal to those of the best known codes of the same length.
In this note we introduce the concept of group convolutional code. We make a complete classification of the minimal $S_3$-convolutional codes over the field of five elements by means of Jategaonkar's theorems.
Split group codes are a class of group algebra codes over an abelian group. They were introduced in 2000 by Ding, Kohel and Ling as a generalization of the cyclic duadic codes. For a prime power q and an abelian group G of order n such that n and q are coprime, consider the group algebra F_{q^2}[G^{*}] of F_{q^2} over the dual group G^{*} of G. We prove that every ideal code in F_{q^{2}}[G^{*}] whose extended code is Hermitian self-dual is a split group code. We characterize the orders of finite abelian groups G for which an ideal code of F_{q^2}[G^{*}] whose extension is Hermitian self-dual e…
Classes of self-dual codes and dual-containing codes are constructed. The codes are obtained within group rings and, using an isomorphism between group rings and matrices, equivalent codes are obtained in matrix form. Distances and other properties are derived by working within the group ring. Quantum codes are constructed from the dual-containing codes.
In this note, an intrinsic description of some families of linear codes with symmetries is given, showing that they can be described more generally as quasi group codes, that is, as linear codes allowing a group of permutation automorphisms which acts freely on the set of coordinates. An algebraic description, including the concatenated structure, of such codes is presented. This allows to construct quasi group codes from codes over rings, and vice versa. The last part of the paper is dedicated to the investigation of self-duality of quasi group codes.
A general method for constructing convolutional codes from units in Laurent series over matrix rings is presented. Using group ring as matrix rings, this forms a basis for in-depth exploration of convolutional codes from group ring encoding, wherein the ring in the group ring is itself a group ring. The method is used to algebraically construct series of convolutional codes. Algebraic methods are used to compute free distances and to construct convolutional codes to prescribed distances.
A structure theorem of the group codes which are relative projective for the subgroup $\lbrace 1 \rbrace$ of $G$ is given. With this, we show that all such relative projective group codes in a fixed group algebra $RG$ are in bijection to the chains of projective group codes of length $\ell$ in the group algebra $\mathbb{F}G$, where $\mathbb{F}$ is the residue field of $R$. We use a given chain to construct the dual code in $RG$ and also derive the minimum Hamming weight as well as a lower bound of the minimum euclidean weight.
