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...

Let $L = U_3(9)$ be the simple projective unitary group in dimension 3 over a field  with 92 elements. In this article, we classify groups with the same order and degree pattern as an almost simple group related to $L$. Since $Aut(L)equiv Z_4$ hence almost simple groups related to $L$ are $L$, $L : 2$ or $L : 4$. In fact, we prove that $L$, $L : 2$ and $L : 4$ are OD-characterizable.

Let $G$ be a finite group and $pi(G)$ be the set of all prime divisors of $|G|$. The prime graph of $G$ is a simple graph $Gamma(G)$ with vertex set $pi(G)$ and two distinct vertices $p$ and $q$ in $pi(G)$ are adjacent by an edge if an only if $G$ has an element of order $pq$. In this case, we write $psim q$. Let $|G= p_1^{alpha_1}cdot p_2^{alpha_2}cdots p_k^{alpha_k}$, where $p_1

Abstract Let $$\mu $$ μ be a log-concave probability measure on $${\mathbb R}^n$$ R n and for any $$N>n$$ N > consider the random polytope $$K_N=\textrm{conv}\{X_1,\ldots ,X_N\}$$ ...

in this paper, we examine that some simple $k_4$-groups can be determined uniquely by their orders and one or two irreducible complex character degrees.

An edge-ordered graph is a with total ordering of its edges. A path $$P=v_1v_2\ldots v_k$$ in an called increasing if $$(v_iv_{i+1}) < (v_{i+1}v_{i+2})$$ for all $$i = 1,\ldots,k-2$$ ; and it decreasing > . We say that P monotone or decreasing. rooted tree T either every from the root to leaf Let G be graph. In straight-line drawing D G, vertices are drawn as different points plane edges straig...

We study a natural question in the Iwasawa theory of algebraic curves genus $$>1$$ . Fix prime number p. Let X be smooth, projective, geometrically irreducible curve defined over field K $$g>1$$ , such that Jacobian has good ordinary reduction at primes above an odd p and for any integer $$n>1$$ let $$K_n^{(p)}$$ denote degree- $$p^n$$ extension contained $$K(\mu _{p^{\infty }})$$ prove explici...

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...

 After the classification of the flag-transitive linear spaces, the attention has been turned to line-transitive linear spaces. In this article, we present a partial classification of the finite linear spaces $mathcal S$ on which an almost simple group $G$ with the socle $G_2(q)$ acts line-transitively.