Minimax Groups Research Articles

Groups with finitely many isomorphism classes of non-modular subgroups

Groups in which the non-moduar subgroups fall into finitely many isomorphism classes are considered, and it is proved that a (generalized) soluble group with this property either has modular subgroup lattice or is a minimax group. The corresponding result for (generalized) soluble groups with finitely many isomorphism classes of non-permutable subgroups is also obtained.

Potency in soluble groups

We prove in particular that if G is a soluble group with no non-trivial locally finite normal subgroups, then G is p-potent for every prime p for which G has no Prüfer p-sections. (A group G is p-potent if for every power n of p and for any element x of G of infinite order or of finite order divisible by n there is a normal subgroup N of G of finite index such that the order of x modulo N is n. A Prüfer p-group is an infinite locally cyclic p-group.) This extends to soluble groups in general, and gives a more direct proof of, recent results of Azarov on polycyclic groups and soluble minimax groups.

On some commutative invariants of modules over minimax nilpotent groups

In the paper we introduce a finite system of invariants for modules over minimax nilpotent groups which consists of classes of equivalent prime ideals of the group algebra of an Abelian minimax group. In particuly, introduced system of invariants allows to study the structure of a minimax nilpotent group N acted by a group of operators G such that N has injective modules which are stabilized by the group of operators G.

An accelerated minimax algorithm for convex-concave saddle point problems with nonsmooth coupling function

In this work we aim to solve a convex-concave saddle point problem, where the convex-concave coupling function is smooth in one variable and nonsmooth in the other and not assumed to be linear in either. The problem is augmented by a nonsmooth regulariser in the smooth component. We propose and investigate a novel algorithm under the name of OGAProx, consisting of an optimistic gradient ascent step in the smooth variable coupled with a proximal step of the regulariser, and which is alternated with a proximal step in the nonsmooth component of the coupling function. We consider the situations convex-concave, convex-strongly concave and strongly convex-strongly concave related to the saddle point problem under investigation. Regarding iterates we obtain (weak) convergence, a convergence rate of order mathcal {O}(frac{1}{K}) and linear convergence like mathcal {O}(theta ^{K}) with theta < 1, respectively. In terms of function values we obtain ergodic convergence rates of order mathcal {O}(frac{1}{K}), mathcal {O}(frac{1}{K^{2}}) and mathcal {O}(theta ^{K}) with theta < 1, respectively. We validate our theoretical considerations on a nonsmooth-linear saddle point problem, the training of multi kernel support vector machines and a classification problem incorporating minimax group fairness.

The Upper And Lower Central Series In Linear Groups

Abstract A classical theorem of Reinhold Baer shows that any group which is finite over its k-th centre has a finite (k + 1)-th term of its lower central series. Although the converse statement is false in general, Philip Hall proved that for any group G the finiteness of γk + 1(G) implies that the index $|G:\zeta_{2k}(G)|$ is finite. Similar situations have been investigated when finiteness is replaced by suitable weaker conditions. Moreover, it was proved by Yuri${\text{\u{\i}}}$ Merzljakov that Baer’s theorem and its potential converse do hold for linear groups. The aim of this paper is to obtain results of the latter type for several other finiteness conditions. Finally, although a result of Baer type does not hold for the class of soluble-by-finite reduced minimax groups, we prove that for this class a theorem of Hall type is true in arbitrary groups.

Two New Species of the Tarsonemus minimax Group (Acari: Heterostigmatina: Tarsonemidae) Associated with Bark Beetles on Quercus falcata in Southeastern United States

Two new species of the Tarsonemus minimax group are described: T. olszanowskii and T. sabelisi. Their morphological descriptions and depictions are presented, and characteristic of their systematic and geographic distribution in the context of the group are discussed. Revised diagnosis and updated key for identification of T. minimax group member-species are included. The current state of knowledge on the group is also summarized.

Groups with all subgroups permutable or soluble of finite rank

In this paper the authors study the class of locally graded groups all of whose subgroups are permutable or are soluble and satisfy a certain rank condition. The rank conditions in question include groups of finite abelian section rank, minimax groups and polycyclic groups. In each case necessary and sufficient conditions are given for a locally graded group to have all subgroups permutable or soluble with the given rank condition.

Virtually torsion-free covers of minimax groups

We prove that every finitely generated, virtually solvable minimax group can be expressed as a homomorphic image of a virtually torsion-free, virtually solvable minimax group. This result enables us to generalize a theorem of Ch. Pittet and L. Saloff-Coste about random walks on finitely generated, virtually solvable minimax groups. Moreover, the paper identifies properties, such as the derived length and the nilpotency class of the Fitting subgroup, that are preserved in the covering process. Finally, we determine exactly which infinitely generated, virtually solvable minimax groups also possess this type of cover.

Right weak Engel conditions on linear groups

If X is a group-class, a group G is right X-Engel if for all g in G there exists an X-subgroup E of G such that for all x in G there is a positive integer m(x) with [g, nx] ∈ E for all n ≥ m(x). Let G be a linear group. Special cases of our main theorem are the following. If X is the class of all Chernikov groups, or all finite groups, or all locally finite groups, then G is right X-Engel if and only if G has a normal X-subgroup modulo which G is hypercentral. The same conclusion holds if G has positive characteristic and X is one of the following classes; all polycyclic-by-finite groups, all groups of finite Prüfer rank, all minimax groups, all groups with finite Hirsch number, all soluble-by-finite groups with finite abelian total rank. In general the characteristic zero case is more complex.

Groups with the Weak Maximal Condition on Non-permutable Subgroups

Let G be a group with the weak maximal condition on non-permutable subgroups. We prove that if G is a generalized radical group then G is either quasihamiltonian or a soluble-by-finite minimax group.

Sylow permutability in soluble minimax groups

A PST-group is a group in which Sylow permutability is a transitive relation. A classification is given of the soluble minimax groups whose finite quotients are PST-groups.

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.

The class of minimax groups is countably recognizable

A class Open image in new window of groups is said to be countably recognizable if a group belongs to Open image in new window whenever all its countable subgroups lie in Open image in new window. It is proved here that the class of minimax groups is countably recognizable.

A Note on the Minimax Solution for the Two-Stage Group Testing Problem

Group testing is an active area of current research and has important applications in medicine, biotechnology, genetics, and product testing. There have been recent advances in design and estimation, but the simple Dorfman procedure introduced by R. Dorfman in 1943 is widely used in practice. In many practical situations, the exact value of the probability p of being affected is unknown. We present both minimax and Bayesian solutions for the group size problem when p is unknown. For unbounded p, we show that the minimax solution for group size is 8, while using a Bayesian strategy with Jeffreys’ prior results in a group size of 13. We also present solutions when p is bounded from above. For the practitioner, we propose strong justification for using a group size of between 8 and 13 when a constraint on p is not incorporated and provide useable code for computing the minimax group size under a constrained p.

On the residual finiteness of the HNN-extensions and generalized free products of finite rank groups

We obtain necessary and sufficient conditions for the residual finiteness for some HNN-extensions and generalized free products of soluble minimax groups.

On the residual finiteness of free products of solvable minimax groups with cyclic amalgamated subgroups

A necessary and sufficient condition for the residual finiteness of a (generalized) free product of two residually finite solvable-by-finite minimax groups with cyclic amalgamated subgroups is obtained. This generalizes the well-known Dyer theorem claiming that every free product of two polycyclic-by-finite groups with cyclic amalgamated subgroups is a residually finite group.

On minimal non- $$M_rC$$ -groups

A group \(G\) is said to be a minimax group if it has a finite series whose factors satisfy either the minimal or the maximal condition. Let \(D(G)\) denotes the subgroup of \(G\) generated by all the Chernikov divisible normal subgroups of \(G\). If \(G\) is a soluble-by-finite minimax group and if \(D(G)=1\) , then \(G\) is said to be a reduced minimax group. Also \(G\) is said to be an \( M_{r}C\)-group (respectively, \(PC\)-group), if \(G/C_{G} \left(x^{G}\right)\) is a reduced minimax (respectively, polycyclic-by-finite) group for all \(x\in G\) . These are generalisations of the familiar property of being an \(FC\)-group. Finally, if \(\mathfrak X \) is a class of groups, then \(G\) is said to be a minimal non-\(\mathfrak X \)-group if it is not an \(\mathfrak X \)-group but all of whose proper subgroups are \(\mathfrak X \)-groups. Belyaev and Sesekin characterized minimal non-\(FC\)-groups when they have a non-trivial finite or abelian factor group. Here we prove that if \(G\) is a group that has a proper subgroup of finite index, then \(G\) is a minimal non-\(M_{r}C\)-group (respectively, non-\(PC\)-group) if, and only if, \(G\) is a minimal non-\(FC\)-group.

A Heuristic Algorithm on Solving the Great Group Dividing of Figure

The problem of the great group of a figure is the famous NP-difficult problem. There exists an algorithm of solving the great group of figure or only applying to some of the special figure .There need time price is index level, and is low efficiency. It puts forward a kind of solving the minimax group partition algorithm with the most magnanimous nodes for elicitation information. This algorithm can be applied to any simple figure, and the maximum time complexity of algorithm is O(sn3).

On modules over group rings of nilpotent groups

We study an R G-module A; where R is a ring, A/C A (G) is not a minimax R-module, C G (A) = 1; and G is a nilpotent group. Let $$ {\mathfrak L} $$ nm (G) be the system of all subgroups H ≤ G such that the quotient modules A/C A (H) are not minimax R-modules. We investigate an R G-module A such that $$ {\mathfrak L} $$ nm (G) satisfies either the weak minimal condition or the weak maximal condition as an ordered set. It is proved that a nilpotent group G satisfying these conditions is a minimax group.

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.