FINITE GROUPS WITH THE SAME JOIN GRAPH AS A FINITE NILPOTENT GROUP

On the planarity of a graph related to the join of subgroups of a finite group

‎Let $G$ be a finite group which is not a cyclic $p$-group‎, ‎$p$ a prime number‎. ‎We define an undirected simple graph $Delta(G)$ whose‎ ‎vertices are the proper subgroups of $G$, which are not contained in the‎ ‎Frattini subgroup of $G$ and two vertices $H$ and $K$ are joined by an edge‎ ‎if and only if $G=langle H‎ , ‎Krangle$‎. ‎In this paper we classify finite groups with planar graph‎. ‎...

Simple groups with the same prime graph as $D_n(q)$

Vasil'ev posed Problem 16.26 in [The Kourovka Notebook: Unsolved Problems in Group Theory, 16th ed.,Sobolev Inst. Math., Novosibirsk (2006).] as follows:Does there exist a positive integer $k$ such that there are no $k$ pairwise nonisomorphicnonabelian finite simple groups with the same graphs of primes? Conjecture: $k = 5$.In [Zvezdina, On nonabelian simple groups having the same prime graph a...

on the planarity of a graph related to the join of subgroups of a finite group

‎let $g$ be a finite group which is not a cyclic $p$-group‎, ‎$p$ a prime number‎. ‎we define an undirected simple graph $delta(g)$ whose‎ ‎vertices are the proper subgroups of $g$, which are not contained in the‎ ‎frattini subgroup of $g$ and two vertices $h$ and $k$ are joined by an edge‎ ‎if and only if $g=langle h‎ , ‎krangle$‎. ‎in this paper we classify finite groups with planar graph‎. ‎...

Finite Groups With a Certain Number of Elements Pairwise Generating a Non-Nilpotent Subgroup

On some Frobenius groups with the same prime graph as the almost simple group ${ {bf PGL(2,49)}}$

The prime graph of a finite group $G$ is denoted by $Gamma(G)$ whose vertex set is $pi(G)$ and two distinct primes $p$ and $q$ are adjacent in $Gamma(G)$, whenever $G$ contains an element with order $pq$. We say that $G$ is unrecognizable by prime graph if there is a finite group $H$ with $Gamma(H)=Gamma(G)$, in while $Hnotcong G$. In this paper, we consider finite groups with the same prime gr...