Quasirecognition by prime graph of the simple group 2 F 4(q)

Abstract

As the main result, we show that if G is a nite group such that Γ(G) = Γ 2 F4(q) , where q = 2 2m+1 for some m 1, then G has a unique nonabelian composition factor isomorphic to 2 F4(q). We also show that if G is a nite group satisfying |G| = 2 F4(q) and Γ(G) = Γ 2 F4(q) , then G ∼ = 2 F4(q). As a consequence of our result we give a new proof for a conjecture of W. Shi and J. Bi for 2 F4(q).

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

“…[11]) и груп-па 2 4 ( ) (см. [12]) квазираспознаваемы по графу простых чисел. Также в статье [13] доказано, что если -простое число, не являющееся простым числом Мерсенна или Ферма, при этом ̸ = 11, 13, 19 и Γ( ) = Γ(PGL(2, )), то имеет единственный неа-белев композиционный фактор, изоморфный PSL(2, ), а в случае = 13 группа имеет единственный неабелев композиционный фактор, изоморфный PSL(2, 13) или PSL (2,27).…”

On the composition factors of a group with the same prime graph as B n (5)

“…In [12] and [18], finite groups with the same prime graph as PSL(2, q), where q is not prime, are determined. It is proved that simple groups F 4 (q), where q = 2 n > 2 (see [10]) and 2 F 4 (q) (see [1]) are quasirecognizable by prime graph. Also in [9], it is proved that if p is a prime number which is not a Mersenne or a Fermat prime and p = 11, 13, 19, and Γ(G) = Γ(PGL(2, p)), then G has a unique nonabelian composition factor which is isomorphic to PSL(2, p); while if p = 13, then G has a unique nonabelian composition factor which is isomorphic to PSL (2,13) or PSL (2,27).…”

CHARACTERIZATION BY PRIME GRAPH OF PGL(2, pk) WHERE p AND k > 1 ARE ODD

“…In [16] and [15], finite groups with the same prime graph as P SL (2, q), where q is not prime, are determined. In [17] and [1], finite groups with the same prime graph as L 16 (2) and 2 F 4 (q), where q = 2 2n+1 > 2 are determined. Also in [13], it is proved that if p is a prime number which is not a Mersenne or Fermat prime and p = 11, 19 and Γ(G) = Γ(PGL(2, p)), then G has a unique nonabelian composition factor which is isomorphic to P SL (2, p) and if p = 13, then G has a unique nonabelian composition factor which is isomorphic to P SL (2,13) or P SL (2,27).…”