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Groups satisfying the two-prime hypothesis with a composition factor isomorphic to PSL 2 ( q ) for q 7

Mark L. Lewis, Yanjun Liu, Hung P. Tong-Viet (2018)

Czechoslovak Mathematical Journal

Let G be a finite group and write cd ( G ) for the degree set of the complex irreducible characters of G . The group G is said to satisfy the two-prime hypothesis if for any distinct degrees a , b cd ( G ) , the total number of (not necessarily different) primes of the greatest common divisor gcd ( a , b ) is at most 2 . We prove an upper bound on the number of irreducible character degrees of a nonsolvable group that has a composition factor isomorphic to PSL 2 ( q ) for q 7 .

OD-characterization of almost simple groups related to L 2 ( 49 )

Liang Cai Zhang, Wu Jie Shi (2008)

Archivum Mathematicum

In the present paper, we classify groups with the same order and degree pattern as an almost simple group related to the projective special linear simple group L 2 ( 49 ) . As a consequence of this result we can give a positive answer to a conjecture of W. J. Shi and J. X. Bi, for all almost simple groups related to L 2 ( 49 ) except L 2 ( 49 ) · 2 2 . Also, we prove that if M is an almost simple group related to L 2 ( 49 ) except L 2 ( 49 ) · 2 2 and G is a finite group such that | G | = | M | and Γ ( G ) = Γ ( M ) , then G M .

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