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Suppose G is a finite group and H is a subgroup of G. H is called weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup $H_{s e}$ of G contained in H such that G = HT and $H \cap T \leq H_{s e}$ ; H is called weakly s-supplemented in G if there is a subgroup T of G such that G = HT and $H \cap T \leq H_{s G}$ , where $H_{s G}$ is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We investigate the influence of the existence of s-permutably embedded and weakly s-supplemented...

Finite Hereditary Near-Field Groups.

Steve Ligh (1978/1979)

Monatshefte für Mathematik

Finite $p$ -nilpotent groups with some subgroups weakly $ℳ$ -supplemented

Liushuan Dong (2020)

Czechoslovak Mathematical Journal

Suppose that $G$ is a finite group and $H$ is a subgroup of $G$ . Subgroup $H$ is said to be weakly $ℳ$ -supplemented in $G$ if there exists a subgroup $B$ of $G$ such that (1) $G = H B$ , and (2) if $H_{1} / H_{G}$ is a maximal subgroup of $H / H_{G}$ , then $H_{1} B = B H_{1} < G$ , where $H_{G}$ is the largest normal subgroup of $G$ contained in $H$ . We fix in every noncyclic Sylow subgroup $P$ of $G$ a subgroup $D$ satisfying $1 < | D | < | P |$ and study the $p$ -nilpotency of $G$ under the assumption that every subgroup $H$ of $P$ with $| H | = | D |$ is weakly $ℳ$ -supplemented in $G$ . Some recent results are generalized.

Finite p'-nilpotent groups. I.

Srinivasan, S. (1987)

International Journal of Mathematics and Mathematical Sciences

Finite p'-nilpotent groups. II.

Srinivasan, S. (1987)

International Journal of Mathematics and Mathematical Sciences

Finite simple groups of Lie type as expanders

Alexander Lubotzky (2011)

Journal of the European Mathematical Society

We prove that all finite simple groups of Lie type, with the exception of the Suzuki groups, can be made into a family of expanders in a uniform way. This confirms a conjecture of Babai, Kantor and Lubotzky from 1989, which has already been proved by Kassabov for sufficiently large rank. The bounded rank case is deduced here from a uniform result for ${S L}_{2}$ which is obtained by combining results of Selberg and Drinfeld via an explicit construction of Ramanujan graphs by Lubotzky, Samuels and Vishne.