International Journal of Group TheoryInternational Journal of Group Theory
http://iranjournals.nlai.ir/1244/
Tue, 25 Feb 2020 03:01:53 +0100FeedCreatorInternational Journal of Group Theory
http://iranjournals.nlai.ir/1244/
Feed provided by International Journal of Group Theory. Click to visit.On noninner automorphisms of finite $p$-groups that fix the center elementwise
http://iranjournals.nlai.ir/1244/article_328225_45660.html
In this paper we show that every finite nonabelian $p$-group $G$ in which the Frattini subgroup $Phi(G)$ has order $leq p^5$ admits a noninner automorphism of order $p$ leaving the center $Z(G)$ elementwise fixed. As a consequence it follows that the order of a possible counterexample to the conjecture of Berkovich is at least $p^8$.Thu, 28 Feb 2019 20:30:00 +0100On the ranks of Fischer group $Fi_{24}^{,prime}$ and the Baby Monster group $mathbb{B}$
http://iranjournals.nlai.ir/1244/article_328226_45660.html
If $G$ is a finite group and $X$ a conjugacy class of elements of $G$, then we define $rank(G{:}X)$ to be the minimum number of elements of $X$ generating $G$. In the present article, we determine the ranks for the Fischer's simple group $Fi_{24}^{prime}$ and the baby monster group $mathbb{B}$.Thu, 28 Feb 2019 20:30:00 +0100Finite groups with the same conjugacy class sizes as a finite simple group
http://iranjournals.nlai.ir/1244/article_328227_45660.html
For a finite group $H$, let $cs(H)$ denote the set of non-trivial conjugacy class sizes of $H$ and $OC(H)$ be the set of the order components of $H$. In this paper, we show that if $S$ is a finite simple group with the disconnected prime graph and $G$ is a finite group such that $cs(S)=cs(G)$, then $|S|=|G/Z(G)|$ and $OC(S)=OC(G/Z(G))$. In particular, we show that for some finite simple group $S$, $G cong S times Z(G)$.Thu, 28 Feb 2019 20:30:00 +0100Finite groups of the same type as Suzuki groups
http://iranjournals.nlai.ir/1244/article_328228_45660.html
For a finite group $G$ and a positive integer $n$, let $G(n)$ be the set of all elements in $G$ such that $x^{n}=1$. The groups $G$ and $H$ are said to be of the same (order) type if $|G(n)|=|H(n)|$, for all $n$. The main aim of this paper is to show that if $G$ is a finite group of the same type as Suzuki groups $Sz(q)$, where $q=2^{2m+1}geq 8$, then $G$ is isomorphic to $Sz(q)$. This addresses to the well-known J. G. Thompson's problem (1987) for simple groups.Thu, 28 Feb 2019 20:30:00 +0100Difference bases in dihedral groups
http://iranjournals.nlai.ir/1244/article_328229_45660.html
A subset $B$ of a group $G$ is called a {em difference basis} of $G$ if each element $gin G$ can be written as the difference $g=ab^{-1}$ of some elements $a,bin B$. The smallest cardinality $|B|$ of a difference basis $Bsubset G$ is called the {em difference size} of $G$ and is denoted by $Delta[G]$. The fraction $eth[G]:=Delta[G]/{sqrt{|G|}}$ is called the {em difference characteristic} of $G$. We prove that for every $nin N$ the dihedral group $D_{2n}$ of order $2n$ has the difference characteristic $sqrt{2}leeth[D_{2n}]leqfrac{48}{sqrt{586}}approx1.983$. Moreover, if $nge 2cdot 10^{15}$, then $eth[D_{2n}]frac{4}{sqrt{6}}approx1.633$. Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality $le80$.Thu, 28 Feb 2019 20:30:00 +0100