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Register a new userOn schurity of finite abelian groups
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A finite group $G$ is called a Schur group, if any Schur ring over $G$ is associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this paper it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any non-cyclic abelian Schur group of odd order is isomorphic to $Z_3times Z_{3^k}$ or $Z_3times Z_3times Z_p$ where $kge 1$ and $p$ is a prime. In addition, we prove that $Z_2times Z_2times Z_p$ is a Schur group for every prime $p$.
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In this article we present an extensive survey on the developments in the theory of non-abelian finite groups with abelian automorphism groups, and pose some problems and further research directions.
A subset $D$ of an Abelian group is $decomposable$ if $emptyset e Dsubset D+D$. In the paper we give partial answer to an open problem asking whether every finite decomposable subset $D$ of an Abelian group contains a non-empty subset $Zsubset D$ with $sum Z=0$. For every $ninmathbb N$ we present a decomposable subset $D$ of cardinality $|D|=n$ in the cyclic group of order $2^n-1$ such that $sum D=0$, but $sum T e 0$ for any proper non-empty subset $Tsubset D$. On the other hand, we prove that every decomposable subset $Dsubsetmathbb R$ of cardinality $|D|le 7$ contains a non-empty subset $Zsubset D$ of cardinality $|Z|lefrac12|D|$ with $sum Z=0$. For every $ninmathbb N$ we present a subset $Dsubsetmathbb Z$ of cardinality $|D|=2n$ such that $sum Z=0$ for some subset $Zsubset D$ of cardinality $|Z|=n$ and $sum T e 0$ for any non-empty subset $Tsubset D$ of cardinality $|T|<n=frac12|D|$. Also we prove that every finite decomposable subset $D$ of an Abelian group contains two non-empty subsets $A,B$ such that $sum A+sum B=0$.
Building on earlier results for regular maps and for orientably regular chiral maps, we classify the non-abelian finite simple groups arising as automorphism groups of maps in each of the 14 Graver-Watkins classes of edge-transitive maps.
Denote by $m(G)$ the largest size of a minimal generating set of a finite group $G$. We estimate $m(G)$ in terms of $sum_{pin pi(G)}d_p(G),$ where we are denoting by $d_p(G)$ the minimal number of generators of a Sylow $p$-subgroup of $G$ and by $pi(G)$ the set of prime numbers dividing the order of $G$.
For a finite group $G$, let $mathrm{diam}(G)$ denote the maximum diameter of a connected Cayley graph of $G$. A well-known conjecture of Babai states that $mathrm{diam}(G)$ is bounded by ${(log_{2} |G|)}^{O(1)}$ in case $G$ is a non-abelian finite simple group. Let $G$ be a finite simple group of Lie type of Lie rank $n$ over the field $F_{q}$. Babais conjecture has been verified in case $n$ is bounded, but it is wide open in case $n$ is unbounded. Recently, Biswas and Yang proved that $mathrm{diam}(G)$ is bounded by $q^{O( n {(log_{2}n + log_{2}q)}^{3})}$. We show that in fact $mathrm{diam}(G) < q^{O(n {(log_{2}n)}^{2})}$ holds. Note that our bound is significantly smaller than the order of $G$ for $n$ large, even if $q$ is large. As an application, we show that more generally $mathrm{diam}(H) < q^{O( n {(log_{2}n)}^{2})}$ holds for any subgroup $H$ of $mathrm{GL}(V)$, where $V$ is a vector space of dimension $n$ defined over the field $F_q$.
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