2019
Том 71
№ 7

# Darafsheh M. R.

Articles: 3
Brief Communications (English)

### Characterization of the group $G_2(5)$ by the prime graph

Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1142-1146

Let $G$ be a finite group. The prime graph of $G$ is a graph $\Gamma (G)$ with vertex set $\pi (G)$ and the set of all prime divisors of $|G|$, where two distinct vertices $p$ and $q$ are adjacent by an edge if $G$ has an element of order $pq$. We prove that if $G\Gamma (G) = \Gamma (G_2(5))$, then $G$ has a normal subgroup $N$ such that $\pi (N) \subseteq \{ 2, 3, 5\}$ and $G/N \sim = G_2(5)$.

Article (English)

### Recognition of the groups $L_5(4)$ and $U_4(4)$ by the prime graph

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 210-217

Let $G$ be a finite group. The prime graph of $G$ is the graph $\Gamma(G)$ whose vertex set is the set $\Pi(G)$ of all prime divisors of the order $|G|$ and two distinct vertices $p$ and $q$ of which are adjacent by an edge if $G$ has an element of order $pq$. We prove that if $S$ denotes one of the simple groups $L_5(4)$ and $U_4(4)$ and if $G$ is a finite group with $\Gamma(G) = \Gamma(S)$, then $G$ has a $G$ normal subgroup $N$ such that $\Pi(N) \subseteq \{2, 3, 5\}$ and $\cfrac GN \cong S$.

Article (English)

### Characterization of $A_{16}$ by a noncommuting graph

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1443–1450

Let $G$ be a finite non-Abelian group. We define a graph $Γ_G$ ; called the noncommuting graph of $G$; with a vertex set $G − Z(G)$ such that two vertices $x$ and $y$ are adjacent if and only if $xy ≠ yx$. Abdollahi, Akbari, and Maimani put forward the following conjecture (the AAM conjecture): If $S$ is a finite non-Abelian simple group and $G$ is a group such that $Γ_S ≅ Γ_G$; then $S ≅ G$. It is still unknown if this conjecture holds for all simple finite groups with connected prime graph except $A_{10}, L_4(8), L_4(4)$, and $U_4(4)$. In this paper, we prove that if $A_{16}$ denotes the alternating group of degree 16; then, for any finite group $G$; the graph isomorphism $Γ_{A_{16}} ≅ Γ_G$ implies that $A_{16} ≅ G$.