相关论文: Rewriting the check of 8-rewritability for $A_5$

An n-ary operation q:A^n->A is called an n-ary quasigroup of order |A| if in x_0=q(x_1,...,x_n) knowledge of any n elements of x_0,...,x_n uniquely specifies the remaining one. An n-ary quasigroup q is permutably reducible if…

Let $G$ be a finite additive abelian group with exponent $n>1$, and let $a_1,\ldots,a_{n-1}\in G$. We show that there is a permutation $\sigma\in S_{n-1}$ such that all the elements $sa_{\sigma(s)}\ (s=1,\ldots,n-1)$ are nonzero if and only…

In this paper we show that a finite nonabelian characteristically simple group G satisfying n = |\pi(G)|+2 if and only if G is isomorphic to A5, where n is the number of isomorphism classes of derived subgroups of G and \pi(G) is the set of…

We investigate the computability of the isomorphism set $\operatorname{Iso}(G_A,G_B)$ between $G_A$ and $G_B$, where $G_A$ is a subgroup of $\mathbb{Q}^n$ generated by columns of integer powers of a non-singular $n \times n$-matrix $A$ with…

Let $G$ be a subgroup of $S_n$, the symmetric group of degree $n$. For any field $k$, $G$ acts naturally on the rational function field $k(x_1,x_2,\ldots,x_n)$ via $k$-automorphisms defined by $\sigma\cdot x_i=x_{\sigma(i)}$ for any…

The quantum permutation group of the set $X_n=\{1,..., n\}$ corresponds to the Hopf algebra $A_{aut}(X_n)$. This is an algebra constructed with generators and relations, known to be isomorphic to $\cc (S_n)$ for $n\leq 3$, and to be…

Let $G$ be a finite group, and $g \in G$. Then $g$ is said to be a vanishing element of $G$, if there exists an irreducible character $\chi$ of $G$ such that $\chi (g)=0$. Denote by ${\rm Vo} (G)$ the set of the orders of vanishing elements…

We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is isomorphic to one of the non-abelian 2-groups of order 8.

A permutation \pi of an abelian group G is said to destroy arithmetic progressions (APs) if, whenever (a,b,c) is a non-trivial 3-term AP in G, that is c-b=b-a and a,b,c are not all equal, then (\pi(a),\pi(b),\pi(c)) is not an AP. In a paper…

Let $G$ be a nonabelian group and $n$ a natural number. We say that $G$ has a strict $n$-split decomposition if it can be partitioned as the disjoint union of an abelian subgroup $A$ and $n$ nonempty subsets $B_1, B_2, \ldots, B_n$, such…

In this paper we study combinatorial aspects of permutations of $\{1,\ldots,n\}$ and related topics. In particular, we prove that there is a unique permutation $\pi$ of $\{1,\ldots,n\}$ such that all the numbers $k+\pi(k)$ ($k=1,\ldots,n$)…

Let $G$ be a finite group and $\mathrm{Irr}(G)$ the set of all irreducible complex characters of $G$. Define the codegree of $\chi \in \mathrm{Irr}(G)$ as $\mathrm{cod}(\chi):=\frac{|G:\mathrm{ker}(\chi) |}{\chi(1)}$ and denote by…

Sets of signed permutation matrices satisfying the GR(4,4) algebra are shown to be, up to sign, left cosets of Klein's famous Vierergruppe. In this way we verify the count done by computer in 2012, and set it in a more significant…

An irreducible character of a finite group $G$ is called quasi $p$-Steinberg character for a prime $p$ if it takes a nonzero value on every $p$-regular element of $G$. In this article, we classify the quasi $p$-Steinberg characters of…

In this paper we survey a new criteria for solvability of finite groups in terms of number of supersolvable (also known as polycyclic) and non-supersolvable subgroups. In particular, we present original examples of supersolvable groups such…

Let \(\nu_p(G)\) be the number of Sylow \(p\)-subgroups of a finite group \(G\), let \(\sigma_p(G)\) be their common order, and set \[ \gamma(G)=\int_0^1\sum_{p\in\pi(G)}\nu_p(G)x^{\sigma_p(G)}\,dx…

An element $g$ in a group $G$ is called reversible if $g$ is conjugate to $g^{-1}$ in $ G $. An element $g$ in $G$ is strongly reversible if $ g $ is conjugate to $g^{-1}$ by an involution in $G$. The group of affine transformations of…

A group is R-harmonious if there exists a permutation $g_1,g_2,\ldots, g_{n-1}$ of the non-identity elements of $G$ such that the consecutive products $g_1g_2$, $g_2g_3$, $\ldots, g_{n-1}g_1$ also form a permutation of the non-identity…

An $n$-ary operation $Q:S^n\to S$ is called an $n$-ary quasigroup of order $|S|$ if in the equation $x_0=Q(x_1,...,x_n)$ knowledge of any $n$ elements of $x_0,...,x_n$ uniquely specifies the remaining one. An $n$-ary quasigroup $Q$ is…

Motivated by a problem from behavioral economics, we study subgroups of permutation groups that have a certain strong symmetry. Given a fixed permutation, consider the set of all permutations with disjoint inversion sets. The group is…