scholarly journals Conjugate purity and infinite groups

1979 ◽  
Vol 28 (1) ◽  
pp. 9-14
Author(s):  
Bola O. Balogun
Keyword(s):  
Finite Group ◽  
Subject Classification ◽  
Infinite Groups ◽  
Soluble Groups ◽  
Infinite Case ◽  
A Finite Group

AbstractIn Balogun (1974), we proved that a finite group in which every subgroup is conjugately pure is necessarily Abelian and we left open the infinite case. In this paper we settle this problem positively for soluble, locally soluble groups and certain classes of groups which include the FC-groups. In the last section of this paper we characterize groups which are conjugately pure in every containing group.Subject classification (Amer. Math. Soc. (MOS) 1970): 20 E 99.

scholarly journals A characterization of finite p-soluble groups of p-length one by commutator identities

1981 ◽  
Vol 30 (3) ◽  
pp. 257-263 ◽  
Author(s):  
Rolf Brandl
Keyword(s):  
Finite Group ◽  
Classical Result ◽  
Soluble Groups ◽  
Similar Description ◽  
A Finite Group ◽  
Engel Condition ◽  
Almost All

AbstractA classical result of M. Zorn states that a finite group is nilpotent if and only if it satisfies an Engel condition. If this is the case, it satisfies almost all Engel conditions. We shall give a similar description of the class of p-soluble groups of p-length one by a sequence of commutator identities.

scholarly journals Finite automorphism groups of laminated near-rings

1983 ◽  
Vol 26 (3) ◽  
pp. 297-306 ◽  
Author(s):  
K. D. Magill ◽  
P. R. Misra ◽  
U. B. Tewari
Keyword(s):  
Finite Group ◽  
Linear Group ◽  
Complex Plane ◽  
Finite Groups ◽  
Automorphism Groups ◽  
The Other ◽  
Complex Polynomial ◽  
Infinite Groups ◽  
Nonsingular Matrices ◽  
A Finite Group

In [3] we initiated our study of the automorphism groups of a certain class of near-rings. Specifically, let P be any complex polynomial and let P denote the near-ring of all continuous selfmaps of the complex plane where addition of functions is pointwise and the product fg of two functions f and g in P is defined by fg=f∘P∘g. The near-ring P is referred to as a laminated near-ring with laminating element P. In [3], we characterised those polynomials P(z)=anzn + an−1zn−1 +…+a0 for which Aut P is a finite group. We are able to show that Aut P is finite if and only if Deg P≧3 and ai ≠ 0 for some i ≠ 0, n. In addition, we were able to completely determine those infinite groups which occur as automorphism groups of the near-rings P. There are exactly three of them. One is GL(2) the full linear group of all real 2×2 nonsingular matrices and the other two are subgroups of GL(2). In this paper, we begin our study of the finite automorphism groups of the near-rings P. We get a result which, in contrast to the situation for the infinite automorphism groups, shows that infinitely many finite groups occur as automorphism groups of the near-rings under consideration. In addition to this and other results, we completely determine Aut P when the coefficients of P are real and Deg P = 3 or 4.

A generalization of a Hall theorem

2016 ◽  
Vol 15 (05) ◽  
pp. 1650085 ◽  
Author(s):  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Chief Factor ◽  
Soluble Groups ◽  
A Finite Group

Let [Formula: see text] be some partition of the set [Formula: see text] of all primes, that is, [Formula: see text] and [Formula: see text] for all [Formula: see text]. We say that a finite group [Formula: see text] is [Formula: see text]-soluble if every chief factor [Formula: see text] of [Formula: see text] is a [Formula: see text]-group for some [Formula: see text]. We give some characterizations of finite [Formula: see text]-soluble groups.

scholarly journals ON A GRAPH RELATED TO THE MAXIMAL SUBGROUPS OF A GROUP

2010 ◽  
Vol 81 (2) ◽  
pp. 317-328 ◽  
Author(s):  
MARCEL HERZOG ◽  
PATRIZIA LONGOBARDI ◽  
MERCEDE MAJ
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Connected Graph ◽  
Solvable Groups ◽  
Maximal Subgroups ◽  
Finite Solvable Group ◽  
Finitely Generated ◽  
Infinite Case ◽  
Locally Graded Group ◽  
A Finite Group

AbstractLet G be a finitely generated group. We investigate the graph ΓM(G), whose vertices are the maximal subgroups of G and where two vertices M1 and M2 are joined by an edge whenever M1∩M2≠1. We show that if G is a finite simple group then the graph ΓM(G) is connected and its diameter is 62 at most. We also show that if G is a finite group, then ΓM(G) either is connected or has at least two vertices and no edges. Finite groups G with a nonconnected graph ΓM(G) are classified. They are all solvable groups, and if G is a finite solvable group with a connected graph ΓM(G), then the diameter of ΓM(G) is at most 2. In the infinite case, we determine the structure of finitely generated infinite nonsimple groups G with a nonconnected graph ΓM(G). In particular, we show that if G is a finitely generated locally graded group with a nonconnected graph ΓM(G), then G must be finite.

Critical maximal subgroups and conjugacy of supplements in finite soluble groups

2018 ◽  
Vol 21 (1) ◽  
pp. 45-63
Author(s):  
Barbara Baumeister ◽  
Gil Kaplan
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Maximal Subgroups ◽  
Abelian Normal Subgroup ◽  
Soluble Groups ◽  
A Finite Group

AbstractLetGbe a finite group with an abelian normal subgroupN. When doesNhave a unique conjugacy class of complements inG? We consider this question with a focus on properties of maximal subgroups. As corollaries we obtain Theorems 1.6 and 1.7 which are closely related to a result by Parker and Rowley on supplements of a nilpotent normal subgroup [3, Theorem 1]. Furthermore, we consider families of maximal subgroups ofGclosed under conjugation whose intersection equals{\Phi(G)}. In particular, we characterize the soluble groups having a unique minimal family with this property (Theorem 2.3, Remark 2.4). In the case when{\Phi(G)=1}, these are exactly the soluble groups in which each abelian normal subgroup has a unique conjugacy class of complements.

scholarly journals On characters in the principal 2-block, II

1979 ◽  
Vol 28 (1) ◽  
pp. 100-106 ◽  
Author(s):  
Marcel Herzog ◽  
Cheryl E. Praeger
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Complex Number ◽  
Irreducible Character ◽  
Subject Classification ◽  
Odd Order ◽  
A Finite Group

AbstractLet k be a non-zero complex number and let u and v be elements of a finite group G. Suppose that at most one of u and v belongs to O(G), the maximal normal subgroup of G of odd order. It is shown that G satisfies X(v)–X(u) = k for every complex nonprincipal irreducible character X in the principal 2-block of G, if and only if G/O(G) is isomorphic to one of the following groups: C2, PSL(2, 2n) or pΣL(2, 52a+1), where n≥2 and a ≥ 1.Subject classification (Amer. Math. Soc. (MOS) 1970): 20 C 20

On residuals of finite groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Stefanos Aivazidis ◽  
Thomas Müller
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Nilpotent Groups ◽  
Simple Groups ◽  
Fitting Subgroup ◽  
Soluble Groups ◽  
A Finite Group ◽  
Fitting Formation ◽  
Original Theorem

Abstract Theorem C in [S. Dolfi, M. Herzog, G. Kaplan and A. Lev, The size of the solvable residual in finite groups, Groups Geom. Dyn. 1 (2007), 4, 401–407] asserts that, in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order and that the inequality is sharp. Inspired by this result and some of the arguments in the above article, we establish the following generalisation: if 𝔛 is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and X ¯ \overline{\mathfrak{X}} is the extension-closure of 𝔛, then there exists an (explicitly known and optimal) constant 𝛾 depending only on 𝔛 such that, for all non-trivial finite groups 𝐺 with trivial 𝔛-radical, | G X ¯ | > | G | γ \lvert G^{\overline{\mathfrak{X}}}\rvert>\lvert G\rvert^{\gamma} , where G X ¯ G^{\overline{\mathfrak{X}}} is the X ¯ \overline{\mathfrak{X}} -residual of 𝐺. When X = N \mathfrak{X}=\mathfrak{N} , the class of finite nilpotent groups, it follows that X ¯ = S \overline{\mathfrak{X}}=\mathfrak{S} , the class of finite soluble groups; thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J. G. Thompson’s classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations 𝔛 of full characteristic such that S ⊂ X ¯ ⊂ E \mathfrak{S}\subset\overline{\mathfrak{X}}\subset\mathfrak{E} , where 𝔈 denotes the class of all finite groups, thus providing applications of our main result beyond the reach of the above theorem.

scholarly journals Formations of Π-soluble groups

1969 ◽  
Vol 10 (1-2) ◽  
pp. 241-250 ◽  
Author(s):  
H. Lausch
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Composition Series ◽  
Soluble Groups ◽  
Hall Subgroups ◽  
A Finite Group

The theory of formations of soluble groups, developed by Gaschütz [4], Carter and Hawkes[1], provides fairly general methods for investigating canonical full conjugate sets of subgroups in finite, soluble groups. Those methods, however, cannot be applied to the class of all finite groups, since strong use was made of the Theorem of Galois on primitive soluble groups. Nevertheless, there is a possiblity to extend the results of the above mentioned papers to the case of Π-soluble groups as defined by Čunihin [2]. A finite group G is called Π-soluble, if, for a given set it of primes, the indices of a composition series of G are either primes belonging to It or they are not divisible by any prime of Π In this paper, we shall frequently use the following result of Čunihin [2]: Ift is a non-empty set of primes, Π′ its complement in the set of all primes, and G is a Π-soluble group, then there always exist Hall Π-subgroups and Hall ′-subgroups, constituting single conjugate sets of subgroups of G respectively, each It-subgroup of G contained in a Hall Π-subgroup of G where each ′-subgroup of G is contained in a Hall Π′-subgroup of G. All groups considered in this paper are assumed to be finite and Π-soluble. A Hall Π-subgroup of a group G will be denoted by G.

scholarly journals A note on subnormality

1976 ◽  
Vol 15 (1) ◽  
pp. 59-64 ◽  
Author(s):  
T.A. Peng
Keyword(s):  
Finite Group ◽  
Minimal Condition ◽  
Soluble Groups ◽  
A Finite Group

Let H be a subgroup of a finite group G and let S be a set of generators of H. We prove that if G is soluble, then H is subnormal in G if and only if there exists an integer n such that for each g in G and a in S the commutator lies in H. This criterion for subnormality is also valid for soluble groups satisfying the maximal or the minimal condition on subgroups.

scholarly journals A note on subnormal defect in finite soluble groups

1989 ◽  
Vol 39 (2) ◽  
pp. 255-258
Author(s):  
R.A. Bryce
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Soluble Groups ◽  
Sylow Subgroups ◽  
Derived Length ◽  
A Finite Group ◽  
Subnormal Subgroups

It is shown that for every positive integer n there exists a finite group of derived length n in which all Sylow subgroups are abeian and in which the defect of subnormal subgroups is at most 3.

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