Finite groups with seminormal Schmidt subgroups

A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Schmidt group. A subgroup A is said to be seminormal in a group G if there exists a subgroup B such that G = AB and AB₁ is a proper subgroup of G, for every proper subgroup B₁ of B. Groups that contain seminormal Schmidt subgroups of even order are considered. In particular, we prove that a finite group is solvable if all Schmidt {2, 3}-subgroups and all 5-closed {2, 5}-Schmidt subgroups of the group are seminormal; the classification of finite groups is not used in so doing. Examples of groups are furnished which show that no one of the requirements imposed on the groups is unnecessary. Supported by BelFBR grant Nos. F05-341 and F06MS-017. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 448–458, July–August, 2007.     

On theta pairs for a maximal subgroup

For a maximal eubgroup M of a finite group G, a 8-pair is any pair of subgroups (C,D) of G such that (i) D?G, D≤C, (ii) - G, - M and (iii) C/D has no proper normal subgroup of G/D. A partial order may be defined on the family of 8-pairs. Let △(M) - {(C,D)|(C,D) is a maximal 8-pair and CM - G}. The purpose of this note is to prove: (1) A group G is solvable if and only if, for each maximal subgroup M of G, △(M) contains a 8-pair (C,D) such that C/D ie nilpctent. (2) If a group G is S₄-free, then G ia eupersolvable if and only if, for each maximal subgroup M of G, △(M) contains a 8-pair (C,D) auch that C/D is cyclic     

真商群为s-自对偶群的有限p-群

如果有限群G的每个子群与G的某个商群同构,则称群G为s-自对偶群.如果s-自对偶群G的每个商群与G的某个子群同构,则称群G为自对偶群.本文分类了每个真商群均为s-自对偶群的有限p-群.作为推论,本文还分类了每个真截段均为s-自对偶群的有限p-群,每个真商群均为自对偶群的有限p-群,以及每个真截段均为自对偶群的有限p-群...     

一类可解的（q）－群

有限群Ｇ叫（ｑ）－群，如果Ｇ中每个次正规子群均为拟正规子群，群Ｇ叫Ｅｑ－群，若Ｇ中每个子群在Ｇ中拟正规或自正规，有限群Ｇ叫内Ｅｑ－群，如果Ｇ本身不是Ｅｑ－群，但Ｇ的每个真子群是Ｅｑ－群，本文确定了Ｅｑ－群的结构与内Ｅｑ－群的分类．     

子群的核平凡或正规闭包极大的有限p群

称有限p群G为ACT群,如果对每个交换子群H,其正规核HG=1或HG=H.又称p群G是CC群,如果对每个非正规交换子群H,有HG=1或HG在G中的指数为p.本文分类了ACT群和CC群.     

The Class Number of Derived Subgroups and the Structure of Camina Groups

A finite group G is called a Camina group if G has a proper normal subgroup N such that gN is precisely a conjugacy class of G for any g ∈E G - N. In this paper, the structure of a Camina group G is determined when N is a union of 2, 3 or 4 conjugacy classes of G.     

Periodic part in some Shunkov groups

A group G is saturated with groups in a set X if every finite subgroup of G is embeddable in G into a subgroup L isomorphic to some group in X. We show that a Shunkov group has a periodic part if the saturating set for it coincides with one of the following: {L₂(q)}, {Sz(q)}, {Re(q)}, or {U₃(2ⁿ)}. Translated fromAlgebra i Logika, Vol. 38, No. 1, pp. 96–125, January–February, 1999.     

Transitive groups with irreducible representations of bounded degree

A well-known theorem of Jordan states that there exists a function J(d) of a positive integer d for which the following holds: if G is a finite group having a faithful linear representation over ℂ of degree d, then G has a normal Abelian subgroup A with [G:A]≤J(d). We show that if G is a transitive permutation group and d is the maximal degree of irreducible representations of G entering its permutation representation, then there exists a normal solvable subgroup A of G such that [G:A]≤J(d) ^log ₂ ^d. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 108–119. Translated by S. A. Evdokimov.     

关于几乎可解群

本文推广了关于局部有限群的Asar定理及p.Hall—Kulatilaka,Kargapolov定理.     

Parabolic subgroups of SL<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> and Sp<Subscript>2<Emphasis Type="Italic">l</Emphasis></Subscript> over a Dedekind ring of arithmetic type

Let R be a commutative ring all of whose proper factor rings are finite and in which a unit of infinite order exists. It is shown that for a subgroup P in G =SL(n, R), n ≥3,or in G =Sp(2l, R), l ≥2, containing the standard Borel subgroup B, the following alternative holds: either P contains a relative elementary subgroup E_I for some ideal I≠0 or H is contained in a proper standard parabolic subgroup. For Dedekind rings of arithmetic type, this allows one, under some mild additional assumptions on units, to completely describe the overgroups of B in G. Bibliography: 30 titles.     

Torsion-free groups with factor-groups on their hypercenter which are periodic

Assume that G is a torsion-free group, Zk(G) is the k-th term of the upper central series of G, and ¯G_k=G/Z_k(G) is a nontrivial periodic group. Then every finite subgroup of ¯G_k is nilpotent of class not higher than k; the group k 2 contains an infinite subgroup with k generators if k2 and two generators if k=1. Moreover any nontrivial invariant subgroup of ¯G_k is infinite. All elements of ¯Gk are of odd order. This assertion is generalized.Translated from Matematicheskie Zametki, Vol. 8, No. 3, pp. 373–383, September, 1970.     

有限群的Deskins极大完备

对于有限群G的一个极大予群M,Deskins称子群C为M的一个完备,如果C(?)M,但C的G-不变真子群包含在M中.令I(M)表示M的所有完备之集. I(M)的一个极大元叫做M的一个极大完备.利用极大完备,本文获得关于群的可解性和超可解性某些新的刻划.     

Nilpotent length of a finite group admitting a frobenius group of automorphisms with fixed-point-free kernel

Suppose that a finite group G admits a Frobenius group FH of automorphisms with kernel F and complement H such that the fixed-point subgroup of F is trivial, i.e., C_G(F) = 1, and the orders of G and H are coprime. It is proved that the nilpotent length of G is equal to the nilpotent length of C_G(H) and the Fitting series of the fixed-point subgroup C_G(H) coincides with a series obtained by taking intersections of C_G(H) with the Fitting series of G.     

无限的可解SD_2－群（Ⅱ）──有限剩余情形

如果对多重循环群Ｇ的每个有限剩余的真子群都是可以由二元生成的，那么我们就把Ｇ叫做ＲＤ＿２－群。在本文里，我们确定了无限的ＲＤ＿２－群的结构，证明了ＲＤ＿２－群是可以由二元生成的。这些结果推广了作者已经得到的关于无限的可解ＳＤ＿２群的全部结果，见［４］．     

Groups with a centralizer of sixth order

Let G be a finite fusion-simple group with a self-centralizing subgroup A of sixth order. It is proved that if the centralizer of the involution from A is an unsolvable subgroup of G of an odd index, then G is isomorphic with the Janko group J₁.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 153–159, July, 1977.The author thanks A. I. Starostin for the formulation of the problem and guidance.     

Complementability conditions for a periodic almost solvable subgroup in the group containing it

It is proved that if every prime Sylow subgroup of a periodic almost solvable (more generally, periodic W₀-) subgroup H of a group G has a complement in G and if, moreover, H is at most countable and the set (H) is finite, the subgroup H itself possesses a complement in G.Deceased.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 6, pp. 822–826, June, 1992.     

极小非PNN-群

有限群G称作PNN-群,如果G的每一个极小子群X要么正规于G,要么是自正规化的.本文将讨论PNN-群的性质,并给出极小非PNN-群,也就是本身不是PNN-群,每个真子群都是PNN-群的有限群的完全分类.     

On θ*-completions of F-abnormal Subgroups

Let F be a saturated formation of finite groups. Given a proper subgroup H of a group G, a subgroup H of G is called F-normal in G if G/HG belongs to F;otherwise H is said to be F-abnormal in G. In this paper, we investigate the structure of a finite group byθ?-completions of F-abnormal subgroups.     

Locally nilpotent groups with a centralizer of finite rank

Locally nilpotent groups in which the centralizer of some finitely generated subgroup has finite rank are studied. It is shown that if G is such a group and F is a finitely generated subgroup with centralizer C_G(F) of finite rank, then the centralizer of the image of F in the factor group G/t(G) modulo the periodic part t(G) also has finite rank. It is also shown that G is hypercentral when F is cyclic and either G is torsion-free or all Sylow subgroups of the periodic part of C_G(F) are finite.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1511–1517, November, 1992.     

On groups with all proper subgroups finite-by-abelian-by-finite

Archiv der Mathematik - We show that if all proper subgroups of a locally graded group G are finite-by-abelian-by-finite, then G contains a finite normal subgroup N such that all proper subgroups...