Related papers: The integer group determinants for $Q_{16}$

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…

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

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 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…

Let ${\rm C}_{4}$ be the cyclic group of order $4$. We determine all possible values of the integer group determinant of ${\rm C}_{4} \rtimes {\rm C}_{4}$.

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 $2$.

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…

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…

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

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

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 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 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…

We show that the integer group determinants for the general affine group of degree one, $GA(1,q)$ with $q=p^k$ a prime power, take the form $D=AB^{q-1},$ where $A$ is a $\mathbb Z_{q-1}$ integer group determinant and $B\equiv A \bmod q$.…

We present the number of totally symmetric quasigroups (equivalently, totally symmetric Latin squares) of order 16, as well as the number of isomorphism classes, and extend previously published results to include information on the number…

Differential calculus on the quantum quaternionic group GL(1,H$_q$) is introduced.

Let $X(Q)=QC$ be a group, where $Q$ is a generalized quaternion group and $C$ is a cyclic group such that $Q\cap C=1$. In this paper, $X(Q)$ will be characterized and moreover, a complete classification for that will be given, provided $C$…

A differential calculus is set up on a deformation of the oscillator algebra. It is uniquely determined by the requirement of invariance under a seven-dimensional quantum group. The quantum space and its associated differential calculus are…