No 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.

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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).

In [J. Algebra 344 (2011), 205–228], a conjecture of J. G. Thompson for PSL