A note on minimal coverings of groups by subgroups

2001 ◽  
Vol 71 (2) ◽  
pp. 159-168 ◽  
Author(s):  
R. A. Bryce ◽  
L. Serena
Keyword(s):  
Maximal Subgroups ◽  
Soluble Groups ◽  
Finite Set ◽  
Group A ◽  
Abelian Subgroups

AbstractA cover for a group is a finite set of subgroups whose union is the whole group. A cover is minimal if its cardinality is minimal. Minimal covers of finite soluble groups are categorised; in particular all but at most one of their members are maximal subgroups. A characterisation is given of groups with minimal covers consisting of abelian subgroups.

Completely simple semigroups: free products, free semigroups and varieties

1981 ◽  
Vol 88 (3-4) ◽  
pp. 293-313 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Matrix Representation ◽  
Maximal Subgroups ◽  
Lattice Of Varieties ◽  
Free Products ◽  
Completely Simple Semigroups ◽  
Countable Rank ◽  
Free Semigroups ◽  
Abelian Subgroups ◽  
And Inversion ◽  
Completely Simple

SynopsisThe class CS of completely simple semigroups forms a variety under the operations of multiplication and inversion (x−1 being the inverse of x in its ℋ-class). We determine a Rees matrix representation of the CS-free product of an arbitrary family of completely simple semigroups and deduce a description of the free completely simple semigroups, whose existence was proved by McAlister in 1968 and whose structure was first given by Clifford in 1979. From this a description of the lattice of varieties of completely simple semigroups is given in terms of certain subgroups of a free group of countable rank. Whilst not providing a “list” of identities on completely simple semigroups it does enable us to deduce, for instance, the description of all varieties of completely simple semigroups with abelian subgroups given by Rasin in 1979. It also enables us to describe the maximal subgroups of the “free” idempotent-generated completely simple semigroups T(α, β) denned by Eberhart et al. in 1973 and to show in general the maximal subgroups of the “V-free” semigroups of this type (which we define) need not be free in any variety of groups.

A Note on the δ-length of Maximal Subgroups in Finite Soluble Groups

1994 ◽  
Vol 166 (1) ◽  
pp. 67-70 ◽  
Author(s):  
A. Ballester-Bolinches ◽  
M. D. Pérez-Ramos
Keyword(s):  
Maximal Subgroups ◽  
Soluble Groups

scholarly journals Schunck classes and projectors in a class of locally finite groups

1995 ◽  
Vol 38 (3) ◽  
pp. 511-522 ◽  
Author(s):  
M. J. Tomkinson
Keyword(s):  
Finite Groups ◽  
Maximal Subgroups ◽  
Locally Finite ◽  
Soluble Groups ◽  
Locally Finite Groups ◽  
Finite Case ◽  
Saturated Formations ◽  
Definition Of ◽  
Schunck Class

We introduce a definition of a Schunck class of periodic abelian-by-finite soluble groups using major subgroups in place of the maximal subgroups used in Finite groups. This allows us to develop the theory as in the finite case proving the existence and conjugacy of projectors. Saturated formations are examples of Schunck classes and we are also able to obtain an infinite version of Gaschütz Ω-subgroups.

SOME CRITERIA FOR EXISTENCE OF SUPPLEMENTS TO NORMAL SUBGROUPS AND THEIR APPLICATIONS

2010 ◽  
Vol 20 (05) ◽  
pp. 689-719 ◽  
Author(s):  
LEONID A. KURDACHENKO ◽  
JAVIER OTAL ◽  
IGOR YA. SUBBOTIN
Keyword(s):  
Normal Subgroups ◽  
Finitely Generated ◽  
Original Proof ◽  
Soluble Groups ◽  
Finite Section ◽  
Abelian Subgroups ◽  
Section Rank ◽  
The Many ◽  
New Criteria

We established several new criteria for existence of complements and supplements to some normal abelian subgroups in groups. In passing, as one of the many useful applications and corollaries of these results, we obtained a description of some finitely generated soluble groups of finite Hirsch–Zaitsev rank. As another application of our results, we obtained a D.J.S. Robinson's theorem on structure of finitely generated soluble groups of finite section rank. The original proof of this theorem was homological, but all proofs in this paper, including this one, are purely group-theoretical.

The Influence of Certain Abelian Subgroups on the Structure of Finite Groups

2008 ◽  
Vol 15 (03) ◽  
pp. 479-484 ◽  
Author(s):  
M. Ramadan
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Prime Power Order ◽  
Group A ◽  
Abelian Subgroups ◽  
A Finite Group

Let G be a finite group. A subgroup K of a group G is called an [Formula: see text]-subgroup of G if NG(K) ∩ Kx ≤ K for all x ∈ G. The set of all [Formula: see text]-subgroups of G is denoted by [Formula: see text]. In this paper, we investigate the structure of a group G under the assumption that certain abelian subgroups of prime power order belong to [Formula: see text].

Isomorphic Subgroups of Finite p-groups Revisited

1974 ◽  
Vol 26 (3) ◽  
pp. 576-579
Author(s):  
William Specht
Keyword(s):  
Finite Group ◽  
Maximal Subgroups ◽  
Group A ◽  
A Finite Group

Several papers of George Glauberman have appeared which analyze the structure of a finite p-group which contains two isomorphic maximal subgroups. The usual setting for an application of these results is a finite group, a p-subgroup, and an isomorphism of this p-group induced by conjugation. In this paper we prove a stronger version of Glauberman's Theorem 8.1 [1].

A note on maximal subgroups of σ-soluble groups

2021 ◽  
pp. 1-5
Author(s):  
Ning Su ◽  
Chenchen Cao ◽  
ShouHong Qiao
Keyword(s):  
Maximal Subgroups ◽  
Soluble Groups

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.

Some sufficient conditions on solvability of finite groups

2016 ◽  
Vol 15 (03) ◽  
pp. 1650057 ◽  
Author(s):  
Wei Meng ◽  
Jiakuan Lu ◽  
Li Ma ◽  
Wanqing Ma
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Conjugacy Classes ◽  
Maximal Subgroups ◽  
Prime Divisors ◽  
Abelian Subgroups ◽  
A Finite Group

For a finite group [Formula: see text], the symbol [Formula: see text] denotes the set of the prime divisors of [Formula: see text] denotes the number of conjugacy classes of maximal subgroups of [Formula: see text]. Let [Formula: see text] denote the number of conjugacy classes of non-abelian subgroups of [Formula: see text] and [Formula: see text] denote the number of conjugacy classes of all non-normal non-abelian subgroups of [Formula: see text]. In this paper, we consider the finite groups with [Formula: see text] or [Formula: see text]. We show these groups are solvable.

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