# On the Solvability of Finite Irreducible Linear Groups with Hall $TI$-Subgroups

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Let G be a finite centerless group, let Ï€( G) be the set of prime divisors of the order of G, and let n( G) be the number of Sylow p-subgroups of G, that is, n( G) = |Syl( G)|. Set NS( G):= { n( G)| p âˆˆ Ï€( G)}. In this paper, we are investigating whether L( r) is determined up to...

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Let â„± be a class of groups and let G be a finite group. We call a set Î£ of subgroups of G a covering subgroup system of G for â„± (or directly an â„±-covering subgroup system of G) if G âˆˆ â„± whenever every subgroup in Î£ is in â„±. We give some covering subgroup...

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Let G be a finite group and let H be a subgroup of G. H is said to be an NR*-subgroup of G if there exists a normal subgroup T of G such that G = HT and if whenever K â— H and g âˆˆ G, then Kg âˆ© H âˆ© T â‰¤ K. A number of new characterizations of a group G are given, under...

- On X-ss-permutable Subgroups of Finite Groups. Feng Peng; Shirong Li; Kun Li; Yanru Bai // Southeast Asian Bulletin of Mathematics;2011, Vol. 35 Issue 2, p285
In this paper, the following concept is introduced: A subgroup H of G is said to be X-ss-permutable in G if there is a nonempty subset X of G and H is X-permutable with all Sylow subgroups of some supplement T of H to G. In this paper, groups with certain X-ss-permutable subgroups of prime power...

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The corepresentation of a Sylow p-subgroup of a symmetric group in the form of generating relations is investigated, and a Sylow subgroup of a group $\rm{S_{p^k}}$, i.e., an n-fold wreath product of regular cyclic groups of prime order, that is isomorphic to the group of automorphisms of a...

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We study Shunkov groups with the following condition: the normalizer of any finite nontrivial subgroup has an almost layer-finite periodic part. Under this condition, we establish the structure of Sylow 2-subgroups in this group.

- Finite Groups with X-permutable Maximal Subgroups of Sylow Subgroups. Jaraden, Jehad Jumah; Al-Dababseh, Awni Faez // Southeast Asian Bulletin of Mathematics;2007, Vol. 31 Issue 6, p1097
Let A, B be subgroups of a group G and ... â‰ X âŠ† G. Then, A is said to be X-permutable with B [6] if there exists an element x âˆˆ X such that ABx = Bx A. In this paper we study finite groups in which maximal subgroups of Sylow subgroups are X-permutable either with maximal...

- On Sylow subgroups of Shunkov periodic groups. Senashov, V. // Ukrainian Mathematical Journal;Nov2005, Vol. 57 Issue 11, p1809
We study the structure of Sylow 2-subgroups in Shunkov periodic groups with almost layer-finite normalizers of finite nontrivial subgroups.

- On weakly Ï„-quasinormal subgroups of finite groups. Lukyanenko, V.; Skiba, A. // Acta Mathematica Hungarica;Nov2009, Vol. 125 Issue 3, p237
Let G be a finite group and H a subgroup of G. We say that: (1) H is Ï„-quasinormal in G if H permutes with all Sylow subgroups Q of G such that (| Q|, | H|) = 1 and (| H|, | Q G|) â‰ 1; (2) H is weakly Ï„-quasinormal in G if G has a subnormal subgroup T such that HT = G and T âˆ©...