Related papers: Minimal Group Determinants For Dicyclic Groups

We prove that all hypergroups of order four are commutative and that there exists a non-comutative hypergroup of order five. These facts imply that the minimum order of non-commutative hypergroups is five even though the minimum order of…

Denote the alternating and symmetric groups of degree $n$ by $A_n$ and $S_n$ respectively. Consider a permutation $\sigma\in S_n$ all of whose nontrivial cycles are of the same length. We find the minimal polynomials of $\sigma$ in the…

The problem of classifying equivalence classes of presentations up to isomorphism of Cayley graphs is considered in this article in the case of dicyclic groups. The number of equivalence classes of presentations is uniformly bounded - it is…

In this note we provide some counterexamples for the conjecture of Moret\'{o} on finite simple groups, which says that any finite simple group $G$ can determined in terms of its order $|G|$ and the number of elements of order $p$, where $p$…

Linear second order recursive sequences with arbitrary initial conditions are studied. For sequences with the same parameters a ring and a group is attached, and isomorphisms and homomorphisms are established for related parameters. In the…

We determine the number of nonequivalent chord diagrams of order $n$ under the action of two groups, $C_{2n}$, a cyclic group of order $2n$, and $D_{2n}$, a dihedral group of order $4n$. Asymptotic formulas are also established.

In this paper, we continue the enumeration of Schur rings over cyclic groups. Cyclic groups of semiprime order $pq$, where $p$ and $q$ are distinct primes, are considered. Additionally, cyclic groups of order $4p$ are considered.

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $D$ be a $p$-divisible group over $k$. Let $n_D$ be the smallest non-negative integer for which the following statement holds: if $C$ is a $p$-divisible group over $k$ of…

We find the nonabelian finite simple groups with order prime divisors not exceeding 1000. More generally, we determine the sets of nonabelian finite simple groups whose maximal order prime divisor is a fixed prime less than 1000. Our…

Given a prime number \(p\) and a natural number \(m\) not divided by \(p\), we propose the problem of finding the smallest number \(r_{0}\) such that for \(r\geq r_{0}\), every group \(G\) of order \(p^{r}m\) has a non-trivial normal…

Let $\mathrm{Diff}_{\partial}(D^{n})$ be the topological group of diffeomorphisms of $D^{n}$ which agree with the identity near the boundary. In this short note, we compute the fundamental groups $\pi_1 \mathrm{Diff}_{\partial}(D^{4k})$ for…

The present paper completes the computation of the separating Noether numbers for the groups with order strictly less than $32$. Most of the results are proved for the case of a general (possibly finite) base field containing an element…

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 investigate which invariants of groups are powerful in distinguishing non-isomorphic p-groups. We introduce the notations of siblings and twins for p-groups that are difficult to distinguish and we describe the siblings and twins among…

In this paper we explicitly determine all indicators for groups isomorphic to the semidirect product of two cyclic groups by an automorphism of prime order, as well as the generalized quaternion groups. We then compute the indicators for…

It is a classical problem to compute a minimal set of invariant polynomial generating the invariant ring of a finite group as an algebra. We present here an algorithm for the computation of minimal generating sets in the non-modular case.…

Given a finite group $G$ of order $n.$ Denote the sum of the inverse-power of element orders in $G$ by $m(G).$ Let $\mathbb{Z}_n$ be the cyclic group of order $n.$ Suppose $G$ is a non-cyclic group of order $n$ then we show that $m(G)\geq…

The minimal faithful permutation degree $\mu(G)$ of a finite group $G$ is the least integer $n$ such that $G$ is isomorphic to a subgroup of the symmetric group $S_n$. If $G$ has a normal subgroup $N$ such that $\mu(G/N) > \mu(G)$, then $G$…

Based on the earlier work of Li (European J. Combin. 1997) and Dobson (Discrete Math. 2008), in this paper we complete the classification of cyclic $m$-DCI-groups and $m$-CI-groups. For a positive integer $m$ such that $m \ge 3$, we show…

We provide polynomial lower bounds for residual finiteness of residually finite, finitely generated solvable groups that admit infinite order elements in the Fitting subgroup of strict distortion at least exponential. For this class of…