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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 .

On a class of finite solvable groups

James Beidleman, Hermann Heineken, Jack Schmidt (2013)

Open Mathematics

A finite solvable group G is called an X-group if the subnormal subgroups of G permute with all the system normalizers of G. It is our purpose here to determine some of the properties of X-groups. Subgroups and quotient groups of X-groups are X-groups. Let M and N be normal subgroups of a group G of relatively prime order. If G/M and G/N are X-groups, then G is also an X-group. Let the nilpotent residual L of G be abelian. Then G is an X-group if and only if G acts by conjugation on L as a group...

On a definition for formations

Marco Barlotti (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

By constructing appropriate faithful simple modules for the group GL(2,3), the author shows that certain "local" definitions for formations are not equivalent.

On a generalization of a theorem of Burnside

Jiangtao Shi (2015)

Czechoslovak Mathematical Journal

A theorem of Burnside asserts that a finite group G is p -nilpotent if for some prime p a Sylow p -subgroup of G lies in the center of its normalizer. In this paper, let G be a finite group and p the smallest prime divisor of | G | , the order of G . Let P ∈ Syl p ( G ) . As a generalization of Burnside’s theorem, it is shown that if every non-cyclic p -subgroup of G is self-normalizing or normal in G then G is solvable. In particular, if P ≇ 〈 a , b | a p n - 1 = 1 , b 2 = 1 , b - 1 a b = a 1 + p n - 2 〉 , where n ≥ 3 for p > 2 and n ≥ 4 for p = 2 , then G is p -nilpotent or p -closed.

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