Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe distinguishing number of quasiprimitive and semiprimitive groups
116
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The distinguishing number of $G leqslant sym(Omega)$ is the smallest size of a partition of $Omega$ such that only the identity of $G$ fixes all the parts of the partition. Extending earlier results of Cameron, Neumann, Saxl and Seress on the distinguishing number of finite primitive groups, we show that all imprimitive quasiprimitive groups have distinguishing number two, and all non-quasiprimitive semiprimitive groups have distinguishing number two, except for $mathrm{GL}(2, 3)$ acting on the eight non-zero vectors of $mathbb F_2^3$, which has distinguishing number three.
rate research
Read More
In this paper we study finite semiprimitive permutation groups, that is, groups in which each normal subgroup is transitive or semiregular. We give bounds on the order, base size, minimal degree, fixity, and chief length of an arbitrary finite semiprimitive group in terms of its degree. To establish these bounds, we classify finite semiprimitive groups that induce the alternating or symmetric group on the set of orbits of an intransitive normal subgroup.
Denote by $ u_p(G)$ the number of Sylow $p$-subgroups of $G$. It is not difficult to see that $ u_p(H)leq u_p(G)$ for $Hleq G$, however $ u_p(H)$ does not divide $ u_p(G)$ in general. In this paper we reduce the question whether $ u_p(H)$ divides $ u_p(G)$ for every $Hleq G$ to almost simple groups. This result substantially generalizes the previous result by G. Navarro and also provides an alternative proof for the Navarro theorem.
Given a finite simplicial graph $Gamma=(V,E)$ with a vertex-labelling $varphi:Vrightarrowleft{text{non-trivial finitely generated groups}right}$, the graph product $G_Gamma$ is the free product of the vertex groups $varphi(v)$ with added relations that imply elements of adjacent vertex groups commute. For a quasi-isometric invariant $mathcal{P}$, we are interested in understanding under which combinatorial conditions on the graph $Gamma$ the graph product $G_Gamma$ has property $mathcal{P}$. In this article our emphasis is on number of ends of a graph product $G_Gamma$. In particular, we obtain a complete characterization of number of ends of a graph product of finitely generated groups.
Given a group $G$, we define the power graph $mathcal{P}(G)$ as follows: the vertices are the elements of $G$ and two vertices $x$ and $y$ are joined by an edge if $langle xranglesubseteq langle yrangle$ or $langle yranglesubseteq langle xrangle$. Obviously the power graph of any group is always connected, because the identity element of the group is adjacent to all other vertices. In the present paper, among other results, we will find the number of spanning trees of the power graph associated with specific finite groups. We also determine, up to isomorphism, the structure of a finite group $G$ whose power graph has exactly $n$ spanning trees, for $n<5^3$. Finally, we show that the alternating group $mathbb{A}_5$ is uniquely determined by tree-number of its power graph among all finite simple groups.
Let $G_{1}$ and $G_{2}$ be disjoint copies of a graph $G$, and let $g:V(G_{1})rightarrow V(G_{2})$ be a function. A functigraph $F_{G}$ consists of the vertex set $V(G_{1})cup V(G_{2})$ and the edge set $E(G_{1})cup E(G_{2})cup {uv:g(u)=v}$. In this paper, we extend the study of the distinguishing number of a graph to its functigraph. We discuss the behavior of the distinguishing number in passing from $G$ to $F_{G}$ and find its sharp lower and upper bounds. We also discuss the distinguishing number of functigraphs of complete graphs and join graphs.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
