Finite Homomorphic Images Research Articles

GROUPS WITH MANY PRONORMAL SUBGROUPS

AbstractA subgroup H of a group G is pronormal in G if each of its conjugates $H^g$ in G is conjugate to it in the subgroup $\langle H,H^g\rangle $ ; a group is prohamiltonian if all of its nonabelian subgroups are pronormal. The aim of the paper is to show that a locally soluble group of (regular) cardinality in which all proper uncountable subgroups are prohamiltonian is prohamiltonian. In order to obtain this result, it is proved that the class of prohamiltonian groups is detectable from the behaviour of countable subgroups. Examples are exhibited to show that there are uncountable prohamiltonian groups that do not behave very well. Finally, it is shown that prohamiltonicity can sometimes be detected through the analysis of the finite homomorphic images of a group.

Almost zip Bezout domain

J. Zelmanowitz introduced the concept of a ring, which we call a zip ring. In this paper we characterize a commutative Bezout domain whose finite homomorphic images are zip rings modulo its nilradical.

Groups with restricted non-permutable subgroups

A subgroup [Formula: see text] of a group [Formula: see text] is said to be permutable if [Formula: see text] for every subgroup [Formula: see text] of [Formula: see text] and the group [Formula: see text] is called metaquasihamiltonian if all subgroups of [Formula: see text] are either permutable or abelian. It is known that a locally graded metaquasihamiltonian group [Formula: see text] is soluble with derived length at most [Formula: see text] and contains a finite normal subgroup [Formula: see text] such that all subgroups of the factor [Formula: see text] are permutable. In this paper, we describe locally graded groups in which the set of all nonmetaquasihamiltonian subgroups satisfies the minimal condition and locally graded groups with the minimal condition on subgroups which are neither abelian nor permutable. Moreover, it is proved here that a finitely generated hyper-(abelian or finite) group whose finite homomorphic images are metaquasihamiltonian is itself metaquasihamiltonian.

Finite Homomorphic Images of Groups of Finite Rank

Let π be a finite set of primes. We prove that each soluble group of finite rank contains a finite index subgroup whose every finite homomorphic π-image is nilpotent. A similar assertion is proved for a finitely generated group of finite rank. These statements are obtained as a consequence of the following result of the article: Each soluble pro-π-group of finite rank has an open normal pronilpotent subgroup.

Finite homomorphic images of groups of finite rank

Finite homomorphic images of groups of finite rank

On residually finite groups of finite general rank

Following A. I.Mal’tsev, we say that a group G has finite general rank if there is a positive integer r such that every finite set of elements of G is contained in some r-generated subgroup. Several known theorems concerning finitely generated residually finite groups are generalized here to the case of residually finite groups of finite general rank. For example, it is proved that the families of all finite homomorphic images of a residually finite group of finite general rank and of the quotient of the group by a nonidentity normal subgroup are different. Special cases of this result are a similar result of Moldavanskii on finitely generated residually finite groups and the following assertion: every residually finite group of finite general rank is Hopfian. This assertion generalizes a similarMal’tsev result on the Hopf property of every finitely generated residually finite group.

Groups whose proper subgroups of infinite rank have minimax conjugacy classes

If [Formula: see text] is a class of groups, then a group [Formula: see text] is said to be a [Formula: see text]-group, if [Formula: see text] is a [Formula: see text]-group for all [Formula: see text]. This is a generalization of the familiar property of being an [Formula: see text]-group. In the present paper we consider a class [Formula: see text] of soluble-by-finite minimax groups such that [Formula: see text] is a subgroup closed class and if [Formula: see text] is a non-[Formula: see text]-group whose proper subgroups of infinite rank are [Formula: see text]-groups, then there exists a prime [Formula: see text] such that every finite homomorphic image of [Formula: see text] is a cyclic [Formula: see text]-group. Our main result states that if [Formula: see text] is a locally (soluble-by-finite) group of infinite rank which has no simple factor group of infinite rank and if all proper subgroups of [Formula: see text] of infinite rank are [Formula: see text]-groups, then so are all proper subgroups of [Formula: see text]. One can take for [Formula: see text] the class of finite, polycyclic-by-finite, Chernikov, reduced minimax or soluble-by-finite minimax groups.

Finite homomorphic images of Bezout duo-domains

It is proved that for a quasi-duo Bezout ring of stable range 1 the duo-ring condition is equivalent to being an elementary divisor ring. As an application of this result a couple of useful properties are obtained for finite homomorphic images of Bezout duo-domains: they are coherent morphic rings, all injective modules over them are flat, their weak global dimension is either 0 or infinity. Moreover, we introduce the notion of square-free element in noncommutative case and it is shown that they are adequate elements of Bezout duo-domains. In addition, we are going to prove that these elements are elements of almost stable range 1, as well as necessary and sufficient conditions for being square-free element are found in terms of regularity, Jacobson semisimplicity, and boundness of weak global dimension of finite homomorphic images of Bezout duo-domains.

A GRAPHICAL TECHNIQUE TO FIND THE PERMUTATION REPRESENTATION OF THE TRIANGLE GROUPS △(2, 3, k)

In this paper we have developed a graphical technique by which a suitably created fragment of a coset diagram for the action of PSL(2, ℤ) on projective lines over Galois fields can be used to obtain a complete coset diagram depicting finite homomorphic images of groups Δ(2, 3, k) = 〈x, y : x2 = y3 = (xy)k = 1〉.

P-groups having few almost-rational irreducible characters

We prove that a 2-group has exactly five rational irreducible characters if and only if it is dihedral, semidihedral or generalized quaternion. For an arbitrary prime p, we say that an irreducible character χ of a p-group G is “almost rational” if ℚ(χ) is contained in the cyclotomic field ℚ p , and we write ar(G) to denote the number of almost-rational irreducible characters of G. For noncyclic p-groups, the two smallest possible values for ar(G) are p 2 and p 2 + p − 1, and we study p-groups G for which ar(G) is one of these two numbers. If ar(G) = p 2 + p − 1, we say that G is “exceptional”. We show that for exceptional groups, |G: G′| = p 2, and so the assertion about 2-groups with which we began follows from this. We show also that for each prime p, there are exceptional p-groups of arbitrarily large order, and for p ≥ 5, there is a pro-p-group with the property that all of its finite homomorphic images of order at least p 3 are exceptional.

Groups Whose Finite Homomorphic Images are Metahamiltonian

A group G is metahamiltonian if all its non-abelian subgroups are normal. It is proved here that a finitely generated soluble group is metahamiltonian if and only if all its finite homomorphic images are metahamiltonian; the behaviour of soluble minimax groups with metahamiltonian finite homomorphic images is also investigated. Moreover, groups satisfying the minimal condition on non-metahamiltonian subgroups are described.

Profinite Heyting Algebras

For a Heyting algebra A, we show that the following conditions are equivalent: (i) A is profinite; (ii) A is finitely approximable, complete, and completely join-prime generated; (iii) A is isomorphic to the Heyting algebra Up(X) of upsets of an image-finite poset X. We also show that A is isomorphic to its profinite completion iff A is finitely approximable, complete, and the kernel of every finite homomorphic image of A is a principal filter of A.

Ideal Structure of the Kauffman and Related Monoids

The generators of the Temperley-Lieb algebra generate a monoid with an appealing geometric representation. It has been much studied, notably by Louis Kauffman. Borisavljević, Došen, and Petrić gave a complete proof of its abstract presentation by generators and relations, and suggested the name “Kauffman monoid”. We bring the theory of semigroups to the study of a certain finite homomorphic image of the Kauffman monoid. We show the homomorphic image (the Jones monoid) to be a combinatorial and regular *-semigroup with linearly ordered ideals. The Kauffman monoid is explicitly described in terms of the Jones monoid and a purely combinatorial numerical function. We use this to describe the ideal structure of the Kauffman monoid and two other of its homomorphic images.

On loops covered by subloops

The topic of this paper is the investigation of coverings of a loop by subloops. A loop has a covering by subloops if it is the set-theoretic union of proper subloops. If the set of subloops is finite, the covering is called finite. Coverings of groups by subgroups have been widely investigated and key results are detailed in the introduction. Various analogues for loops of the results for groups are obtained. An example of an infinite loop which is the union of three proper commutative subloops, but has no finite homomorphic images and has a trivial center, shows that the results for loops cannot be as general as for groups, justifying additional assumptions on the loops or the coverings.

Polycyclic groups with modular finite homomorphic images

A group G is said to be a modular group if it has modular subgroup lattice. We will prove in this paper that a polycyclic group G is modular if and only if all its finite homomorphic images are modular groups. Similar results will also be obtained for other conditions of modular type.

Finite images and characters of topological groups

We investigate conditions under which every finite homomorphic image of a topological group is obtained as that of a continuous homomorphism. An Abelian topological group has such property if it is compact or if it has bounded exponent and sufficiently many characters.

Conjugacy Separability of Certain HNN Extensions of Conjugacy-Separable Groups

A group G is said to be conjugacy-separable if, for each pair of elements x, y ɛ G such that x and y are not conjugate in G, there exists a finite homomorphic image G¯ of G such that the images of x and y are not conjugate in G¯. In this paper, we show that certain HNN extensions of conjugacy-separable groups are conjugacy-separable. We then apply our results to HNN extensions of polycyclic-by-finite groups.

Conjugacy Separability of Certain HNN Extensions of Conjugacy-Separable Groups

A group G is said to be conjugacy-separable if, for each pair of elements x, y ɛ G such that x and y are not conjugate in G, there exists a finite homomorphic image G¯ of G such that the images of x and y are not conjugate in G¯. In this paper, we show that certain HNN extensions of conjugacy-separable groups are conjugacy-separable. We then apply our results to HNN extensions of polycyclic-by-finite groups.

Tree Products of Conjugacy Separable Groups

A group G is said to be conjugacy separable if for each pair of elements x, y∈G such that x and y are not conjugate in G, there exists a finite homomorphic image Ḡ of G such that the images of x, y are not conjugate in Ḡ. In this note, we show that the tree products of finitely many conjugacy separable and subgroup separable groups amalgamating central subgroups with trivial intersections are conjugacy separable. We then apply our results to polycyclic-by-finite groups and free-by-finite groups.

On Finite Homomorphic Images of the Multiplicative Group of a Division Algebra

Conjecture 2 (G. Margulis and V. Platonov). Let G be a simple, simply connected algebraic group defined over an algebraic number field K. Let T be the set of all nonarchimedean places v of K such that G is Kv-anisotropic; then for any noncentral normal subgroup N ≤ G(K) there exists an open normal subgroup W ≤ G(K,T ) = ∏ v∈T G(Kv) such that N = G(K) ∩ W ; in particular, if T = ∅ then G(K) does not have proper noncentral normal subgroups.