Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGraphs encoding the generating properties of a finite group
188
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Assume that $G$ is a finite group. For every $a, b inmathbb N,$ we define a graph $Gamma_{a,b}(G)$ whose vertices correspond to the elements of $G^acup G^b$ and in which two tuples $(x_1,dots,x_a)$ and $(y_1,dots,y_b)$ are adjacent if and only if $langle x_1,dots,x_a,y_1,dots,y_b rangle =G.$ We study several properties of these graphs (isolated vertices, loops, connectivity, diameter of the connected components) and we investigate the relations between their properties and the group structure, with the aim of understanding which information about $G$ are encoded by these graphs.
rate research
Read More
A group $G$ is said to be $frac{3}{2}$-generated if every nontrivial element belongs to a generating pair. It is easy to see that if $G$ has this property then every proper quotient of $G$ is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if $G$ is a finite group and every proper quotient of $G$ is cyclic, then for any pair of nontrivial elements $x_1,x_2 in G$, there exists $y in G$ such that $G = langle x_1, y rangle = langle x_2, y rangle$. In other words, $s(G) geqslant 2$, where $s(G)$ is the spread of $G$. Moreover, if $u(G)$ denotes the more restrictive uniform spread of $G$, then we can completely characterise the finite groups $G$ with $u(G) = 0$ and $u(G)=1$. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups whose socles are exceptional groups of Lie type and this is the case we treat in this paper.
In this paper we have investigated some properties of the power graph and commuting graph associated with a finite group, using their tree-numbers. Among other things, it has been shown that the simple group $L_2(7)$ can be characterized through the tree-number of its power graph. Moreover, the classification of groups with power-free decomposition is presented. Finally, we have obtained an explicit formula concerning the tree-number of commuting graphs associated with the Suzuki simple groups.
We study the class of finite groups $G$ satisfying $Phi (G/N)= Phi(G)N/N$ for all normal subgroups $N$ of $G$. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe to all finite groups and answer in the affirmative a long-standing question of Christensen whether the class of finite groups which possess complements for each of their normal subgroups is subnormally closed.
A {it $k$-involution} is an involution with a fixed point set of codimension $k$. The conjugacy class of such an involution, denoted $S_k$, generates $text{Mob}(n)$-the the group of isometries of hyperbolic $n$-space-if $k$ is odd, and its orientation preserving subgroup if $k$ is even. In this paper, we supply effective lower and upper bounds for the $S_k$ word length of $text{Mob}(n)$ if $k$ is odd, and the $S_k$ word length of $text{Mob}^+(n)$, if $k$ is even. As a consequence, for a fixed codimension $k$ the length of $text{Mob}^{+}(n)$ with respect to $S_k$, $k$ even, grows linearly with $n$ with the same statement holding in the odd case. Moreover, the percentage of involution conjugacy classes for which $text{Mob}^{+}(n)$ has length two approaches zero, as $n$ approaches infinity.
For a finite group $G$ with a normal subgroup $H$, the enhanced quotient graph of $G/H$, denoted by $mathcal{G}_{H}(G),$ is the graph with vertex set $V=(Gbackslash H)cup {e}$ and two vertices $x$ and $y$ are edge connected if $xH = yH$ or $xH,yHin langle zHrangle$ for some $zin G$. In this article, we characterize the enhanced quotient graph of $G/H$. The graph $mathcal{G}_{H}(G)$ is complete if and only if $G/H$ is cyclic, and $mathcal{G}_{H}(G)$ is Eulerian if and only if $|G/H|$ is odd. We show some relation between the graph $mathcal{G}_{H}(G)$ and the enhanced power graph $mathcal{G}(G/H)$ that was introduced by Sudip Bera and A.K. Bhuniya (2016). The graph $mathcal{G}_H(G)$ is complete if and only if $G/H$ is cyclic if and only if $mathcal{G}(G/H)$ is complete. The graph $mathcal{G}_H(G)$ is Eulerian if and only if $|G|$ is odd if and only if $mathcal{G}(G)$ is Eulerian, i.e., the property of being Eulerian does not depend on the normal subgroup $H$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
