Quasirecognition by prime graph of finite simple groups L n (2) and U n (2)

Abstract

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p are joined by an edge if G has an element of order pp . Let L = Ln(2) or Un (2), where n 17. We prove that L is quasirecognizable by prime graph, i.e. if G is a finite group such that Γ(G) = Γ(L), then G has a unique nonabelian composition factor isomorphic to L. As a consequence of our result we give a new proof for the recognition by element orders of Ln(2). Also we conclude that the simple group Un(2) is quasirecognizable by element orders.

Квазираспознаваемость $^2D_{n}(3^\alpha)$ по графу простых чисел при $n=4m+1\ge 21$ и нечетном $\alpha$

“…Далее, в [14] доказано, что если и > 1 нечетны и = -простая степень, то PGL(2, ) единственным образом определяется своим графом простых чисел. В [15]- [18] получены конечные группы с таким же графом простых чисел, как у (2). А в работе [19] доказано, что некоторые ортогональные группы являются квазираспознаваемыми по графу простых чисел.…”

Quasirecognition by Prime Graph of the Groups 2D2n(q) Where q < 105

“…Quasirecognizability by prime graph of groups G 2 (3 2n+1 ) and 2 B 2 (2 2n+1 ) has been proved in [4]. In [5][6][7], finite groups with the same prime graphs as Γ [8], it is proved that if p is a prime less than 1000, for suitable n, the finite simple groups L n (p) and U n (p) are quasirecognizable by prime graph. Now as the main result of this paper, we prove the following theorem:…”

Quasirecognition by Prime Graph of the Groups ²<em>D</em>_2<em>n</em>(<em>q</em>) Where <em>q</em> &lt; 10⁵

“…In [7,8,9] finite groups with the same prime graph as Γ(L n (2)), Γ(U n (2)), Γ(D n (2)), Γ( 2 D n (2)) and Γ( 2 D 2k (3)) are obtained. Also in [10] it is proved that if p is a prime less than 1000 and for suitable n, the finite sinple groups L n (p), U n (p) are quasirecognizable by prime graph.…”