Related papers: Integer group determinants for ${\rm C}_{4} \rtime…

We obtain a complete description of the integer group determinants for SmallGroup(16,8), the semidihedral group of order 16. While this paper was in preparation, a complete descriptions for this group was independently obtained by Yuka…

The exact degree bound for the generators of rings of polynomial invariants is determined for the finite, non-cyclic groups having a cyclic subgroup of index two. It is proved that the Noether number of these groups equals one half the…

We establish a congruence satisfied by the integer group determinants for the non-abelian Heisenberg group of order $p^3$. We characterize all determinant values coprime to $p$, give sharp divisibility conditions for multiples of $p$, and…

Olga Taussky-Todd suggested the problem of determining the possible values of integer circulant determinants. To solve a special case of the problem, Laquer gave a factorization of circulant determinants. In this paper, we give a modest…

In this paper, we generalise the formula for the fourth moment of a random determinant to account for entries with asymmetric distribution. We also derive the second moment of a random Gram determinant.

We prove a conjecture of Dukes and Herke concerning the possible orders of a basis for the cyclic group Z_n, namely : For each k \in N there exists a constant c_k > 0 such that, for all n \in N, if A \subseteq Z_n is a basis of order…

An integral of a group $G$ is a group $H$ whose commutator subgroup is isomorphic to $G$. In this paper, we prove that the integrability of a finite group is a decidable problem.

This paper is devoted to the analysis of a false generalization of the rule of Sarrus and its properties that can be derived with the help of dihedral groups. Further, we discuss a Sarrus-like scheme that could be helpful for students to…

In this paper, cyclic codes are established over some finite quaternion integer rings with respect to the quaternion Mannheim distance, and de- coding algorithm for these codes is given.

The alpha-determinant unifies and interpolates the notion of the determinant and permanent. We determine the irreducible decomposition of the cyclic module of $gl_n(C)$ defined by the alpha-determinant. The degeneracy of the irreducible…

We give a formula for the determinant of an $n\times n$ matrix with entries from a commutative ring with unit. The formula can be evaluated by a "straight-line program" performing only additions, subtractions and multiplications of ring…

We are interested in ordering the elements of a subset A of the non-zero integers modulo n in such a way that all the partial sums are distinct. We conjecture that this can always be done and we prove various partial results about this…

Let $G$ be a finite group, $L_1(G)$ be its poset of cyclic subgroups and consider the quantity $\alpha(G)=\frac{|L_1(G)|}{|G|}$. The aim of this paper is to study the class $\cal{C}$ of finite nilpotent groups having…

Let $N(n)$ denote the number of isomorphism types of groups of order $n$. We consider the integers $n$ that are products of at most $4$ not necessarily distinct primes and exhibit formulas for $N(n)$ for such $n$.

Any Schur ring is uniquely determined by a partition of the elements of the group. An open question in the study of Schur rings is determining which partitions of the group induce a Schur ring. Although a structure theorem is available for…

In this paper we introduce and study the concept of cyclic factorization number of a finite group G. By using the Mobius inversion formula and other methods involving the cyclic subgroup structure, this is explicitly computed for some…

Cappell's unitary nilpotent groups UNil(R;R,R) are calculated for the integral group ring R=Z[C_2] of the cyclic group C_2 of order two. Specifically, they are determined as modules over the Verschiebung algebra V using the…

Two known computation methods and one new computation method for matrix determinant over an integral domain are discussed. For each of the methods we evaluate the computation times for different rings and show that the new method is the…

We present a deterministic polynomial-time algorithm that determines whether a finite module over a finite commutative ring is cyclic, and if it is, outputs a generator.

A ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive code ${\cal C}\subseteq{\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta$ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb{Z}}_2$ and the set of…