scholarly journals Supplementation in groups

2000 ◽  
Vol 42 (1) ◽  
pp. 37-50 ◽  
Author(s):  
Luise-Charlotte Kappe ◽  
Joseph Kirtland
Keyword(s):  
Subject Classification ◽  
Characteristic Subgroup ◽  
Normal Subgroups ◽  
Closure Properties

In this paper, groups are investigated in which all subgroups, all normal subgroups, or all characteristic subgroups have a proper supplement. This supplement can be either an arbitrary subgroup, a normal or a characteristic subgroup, resulting in nine classes of groups. Properties of these classes are studied such as containment and closure properties, and characterizations for several of these classes are given.1991 Mathematics Subject Classification Primary 20E34, Secondary 20E15.

scholarly journals Existentially closed locally cofinite groups

1992 ◽  
Vol 35 (2) ◽  
pp. 233-253 ◽  
Author(s):  
Felix Leinen
Keyword(s):  
Topological Group ◽  
Finite Groups ◽  
Fundamental System ◽  
Study Groups ◽  
Normal Subgroups ◽  
Closure Properties ◽  
Existentially Closed ◽  
Trivial Intersection ◽  
Direct Limits

Let be a class of finite groups. Then a c-group shall be a topological group which has a fundamental system of open neighbourhoods of the identity consisting of normal subgroups with -factor groups and trivial intersection. In this note we study groups which are existentially closed (e.c.) with respect to the class Lc of all direct limits of c-groups (where satisfies certain closure properties). We show that the so-called locally closed normal subgroups of an e.c. Lc-group are totally ordered via inclusion. Moreover it turns out that every ∀2-sentence, which is true for countable e.c. L-groups, also holds for e.c. Lc-groups. This allows it to transfer many known properties from e.c. L-groups to e.c. Lc-groups.

scholarly journals Soluble groups in which every finitely generated subgroup is finitely presented

1978 ◽  
Vol 26 (1) ◽  
pp. 115-125 ◽  
Author(s):  
J. R. J. Groves
Keyword(s):  
Subject Classification ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Maximal Condition ◽  
Soluble Groups ◽  
Image Position ◽  
Finitely Presented

AbstractThe class of finitely generated soluble coherent groups is considered. It is shown that these groups have the maximal condition on normal subgroups and can be characterized in a number of ways. In particular, they are precisely the class of finitely generated soluble groups G with the property:Subject classification (Amer. Math. Soc. (MOS) 1970): primary 20 E 15; secondary 20 F 05.

scholarly journals An extension of the Glauberman ZJ-theorem

2020 ◽  
pp. 1-17
Author(s):  
M. Yasi̇r Kızmaz
Keyword(s):  
Free Group ◽  
Closed Subgroup ◽  
Characteristic Subgroup ◽  
Normal Subgroups ◽  
Stable Group

Let [Formula: see text] be an odd prime and let [Formula: see text], [Formula: see text] and [Formula: see text] denote the three different versions of Thompson subgroups for a [Formula: see text]-group [Formula: see text]. In this paper, we first prove an extension of Glauberman’s replacement theorem [G. Glauberman, A characteristic subgroup of a p-stable group, Canad. J. Math. 20 (1968) 1101–1135, Theorem 4.1]. Second, we prove the following: Let [Formula: see text] be a [Formula: see text]-stable group and [Formula: see text]. Suppose that [Formula: see text]. If [Formula: see text] is a strongly closed subgroup in [Formula: see text], then [Formula: see text], [Formula: see text] and [Formula: see text] are normal subgroups of [Formula: see text]. Third, we show the following: Let [Formula: see text] be a [Formula: see text]-free group and [Formula: see text]. If [Formula: see text] is a strongly closed subgroup in [Formula: see text], then the normalizers of the subgroups [Formula: see text], [Formula: see text] and [Formula: see text] control strong [Formula: see text]-fusion in [Formula: see text]. We also prove a similar result for a [Formula: see text]-stable and [Formula: see text]-constrained group. Finally, we give a [Formula: see text]-nilpotency criteria, which is an extension of Glauberman–Thompson [Formula: see text]-nilpotency theorem.

A characteristically simple group

1954 ◽  
Vol 50 (4) ◽  
pp. 641-642 ◽  
Author(s):  
D. H. McLain
Keyword(s):  
Simple Group ◽  
Finite Order ◽  
Commutator Subgroup ◽  
Unit Group ◽  
Characteristic Subgroup ◽  
Normal Subgroups ◽  
Locally Finite ◽  
Finite Set ◽  
Group A ◽  
Trivial Centre

The object of this note is to give an example of an infinite locally finite p-group which has no proper characteristic subgroup except the unit group. (A group G is a locally finite p-group if every finite set of elements of G generates a subgroup of finite order equal to a power of the prime p.) It is known that an infinite locally finite p-group cannot be simple, for if it were it would satisfy the minimal condition for normal subgroups, and so have a non-trivial centre (see(1)). However our example shows that it can be characteristically-simple. Examples are known of locally finite p-groups with trivial centre ((2), (4)), and of locally finite p-groups coinciding with their commutator groups ((1), (5)). Since the centre and commutator subgroup of a group are characteristic subgroups our example will have both of these properties. We may remark that the direct product of a simple, or even of a characteristically-simple group with itself any number of times is also characteristically-simple, but by Corollary 2.1 our group cannot be so decomposed.

Groups of Wielandt length two

1991 ◽  
Vol 110 (2) ◽  
pp. 229-244 ◽  
Author(s):  
Elizabeth A. Ormerod
Keyword(s):  
Finite Group ◽  
Nilpotent Groups ◽  
Subnormal Subgroup ◽  
Characteristic Subgroup ◽  
Normal Subgroups ◽  
Trivial Group ◽  
A Finite Group

The Wielandt subgroup ω(G) of a group G is the subgroup of elements that normalize every subnormal subgroup of G. This subgroup, now named for Wielandt, was introduced by him in 1958 [15]. For a finite non-trivial group the Wielandt subgroup is always a non-trivial, characteristic subgroup. Thus it is possible to define the ascending Wielandt series for a finite group G which terminates at the group. Write ω0(G) = 1, and for i ≥ 1, ωi(G)/ωi–1(G) = ω(G/ωi–1(G)). The smallest n such that ωn(G) = G is called the Wielandt length of G, and the class of groups of Wielandt length at most n is denoted by . From the definition it follows that is closed under homomorphic images and taking normal subgroups. Nilpotent groups in are also closed under taking subgroups.

Classical Tauberian Theorems for the (C; 1; 1) Summability Method

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Ümit Totur
Keyword(s):  
Tauberian Theorem ◽  
Summability Method ◽  
Subject Classification ◽  
Tauberian Theorems ◽  
Double Sequences ◽  
Littlewood Theorem

Abstract In this paper we generalize some classical Tauberian theorems for single sequences to double sequences. One-sided Tauberian theorem and generalized Littlewood theorem for (C; 1; 1) summability method are given as corollaries of the main results. Mathematics Subject Classification 2010: 40E05, 40G0

Some Properties of Para-Sasakian Manifolds

2017 ◽  
Vol 5 (2) ◽  
pp. 73-78
Author(s):  
Jay Prakash Singh ◽  
Keyword(s):  
Vector Field ◽  
Killing Vector ◽  
Sasakian Manifold ◽  
Subject Classification ◽  
Killing Vector Field ◽  
Sasakian Manifolds ◽  
Geometric Properties ◽  
Paper Author

In this paper author present an investigation of some differential geometric properties of Para-Sasakian manifolds. Condition for a vector field to be Killing vector field in Para-Sasakian manifold is obtained. Mathematics Subject Classification (2010). 53B20, 53C15.

Multiple Regularity and Binary ETOL-Systems1

1981 ◽  
Vol 4 (1) ◽  
pp. 19-34
Author(s):  
Ryszard Danecki
Keyword(s):  
Equivalence Problem ◽  
Tree Automata ◽  
Closure Properties ◽  
Multiple Tree

Closure properties of binary ETOL-languages are investigated by means of multiple tree automata. Decidability of the equivalence problem of deterministic binary ETOL-systems is proved.

On metalinear ETOL systems

1980 ◽  
Vol 3 (1) ◽  
pp. 15-36
Author(s):  
Grzegorz Rozenberg ◽  
Dirk Vermeir
Keyword(s):  
Finite Index ◽  
Closure Properties ◽  
Pumping Lemma ◽  
L Systems ◽  
Structural Characterizations

The concept of metalinearity in ETOL systems is investigated. Some structural characterizations, a pumping lemma and the closure properties of the resulting class of languages are established. Finally, some applications in the theory of L systems of finite index are provided.

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