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

We determine all possible values of the integer group determinant of ${\rm C}_{4}^{2}$, where ${\rm C}_{4}$ is the cyclic group of order $4$.

We determine all possible values of the integer group determinant of ${\rm C}_{2}^{4}$, where ${\rm C}_{2}$ is the cyclic group of order $2$.

For any positive integer $n$, let ${\rm C}_{n}$ be the cyclic group of order $n$. We determine all possible values of the integer group determinant of ${\rm C}_{2}^{2} \rtimes C_{4}$.

For any positive integer $n$, let ${\rm C}_{n}$ be the cyclic group of order $n$. We determine all possible values of the integer group determinant of ${\rm C}_{4} \times {\rm C}_{2}^{2}$, which is the only unsolved abelian group of order…

For the symmetric group $S_4$ we determine all the integer values taken by its group determinant when the matrix entries are integers.

For every group of order at most 14 we determine the values taken by its group determinant when its variables are integers.

We determine the minimal non-trivial integer group determinant for the dicyclic group of order $4n$ when $n$ is odd. We also discuss the set of all integer group determinants for the dicyclic groups of order $4p$.

We obtain a complete description of the integer group determinants for $Q_{16},$ the dicyclic or generalized quaternion group of order 16.

Let $\mathbb Z_n$ denote the cyclic group of order $n$. We show how the group determinant for $G= \mathbb Z_n \times H$ can be simply written in terms of the group determinant for $H$. We use this to get a complete description of the…

We obtain a complete description of the integer group determinants for SmallGroup(16,13), the central product of the dihedral group of order eight and cyclic group of order four. These values are the same as the integer group determinants…

We give a necessary and sufficient condition for a prime to be an integer group determinant for an arbitrary abelian $p$-group of the form ${\rm C}_{p} \times H$, where ${\rm C}_{p}$ is the cyclic group of order $p$. Also, we show that…

In this paper, we give a refinement of a generalized Dedekind's theorem. In addition, we show that all possible values of integer group determinants of any group are also possible values of integer group determinants of its any abelian…

We obtain a complete description of the integer group determinants for $\mathbb Z_{18}$ (these are the $18\times18$ circulant determinants with integer entries) and $\mathbb Z_3 \times \mathbb Z_6$, the two abelian groups of order 18. This…

We obtain a complete description of the integer group determinants for the non-abelian groups of order 18.

For a 4th order 3-dimensional cyclic symmetric tensor, a sufficient and necessary condition is bulit for its positive semi-definiteness. A sufficient and necessary condition of positive definiteness is showed for a 4th order $n$-dimensional…

We consider indecomposable representations of the Klein four group over a field of characteristic $2$ and of a cyclic group of order $pm$ with $p,m$ coprime over a field of characteristic $p$. For each representation we explicitly describe…

Taking the absolute value of consecutive differences of a cyclicly ordered list of integers constitutes a simple dynamical system. For lists of lenght a power of two the process will terminate in all zeros, but examples with arbitarily long…

We consider the values taken by $n\times n$ circulant determinants with integer entries when $n$ is the product of two distinct odd primes $p,q$. These correspond to the integer group determinants for $\mathbb Z_{pq}$, the cyclic group of…

In the following short paper we list some useful results concerning determinants and inverses of matrices. First we show, how to calculate determinants of $d \times d$ matrices, if their traces are known. As a next step $4 \times 4$…

Let KG be a group algebra of a finite p-group G over a finite field K of characteristic p. We compute the order of the unitary subgroup of the group of units when G is either an extraspecial 2-group or the central product of such a group…