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

Abstract

Int. J. Algebra Comput. 2010.20:847-873. Downloaded from www.worldscientific.com by UNIVERSITY OF SEVILLE on 02/05/15. For personal use only. Characterization by Prime Graph of PGL(2, p k ) 849 divisor of positive integers a and b, and by [a, b] the least common multiple of positive integers a and b. Let m be a positive integer and p be a prime number.Then |m| p denotes the p-part of m. In other words, |m| p = p k if p k | m but p k+1 m. Preliminary ResultsRemark 2.

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

“…Therefore these simple groups are quasirecognizable by element orders. As a consequence of our result we give a new proof for the recognition by element orders of L n (2).…”

“…In [1,14], finite groups with the same prime graph as 2 F 4 (q), where q = 2 2n+1 > 2, and F 4 (q) where q = 2 n > 2 are determined. Also in [9], it is proved that if p is a prime number which is not a Mersenne or Fermat prime and p = 11, 13, 19 and Γ(G) = Γ P GL (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). Then it is proved that if p and k > 1 are odd and q = p k is a prime power, then P GL (2, q) is uniquely determined by its prime graph [2] (see also [4,7]).…”

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

“…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). 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).…”

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

“…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). Then it is proved that for an odd prime p and odd k > 2, PGL(2, p k ) is recognizable by its prime graph [2].…”

“…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). Then it is proved that for an odd prime p and odd k > 2, PGL(2, p k ) is recognizable by its prime graph [2]. In [15], [16], [17], [19] finite groups with the same prime graph as L n (2) are obtained.…”