Minimal Number of Generators and Minimum Order of a Non-Abelian Group Whose Elements Commute with Their Endomorphic Images

A group in which every element commutes with its endomorphic images is called an “E-group″. If p is a prime number, a p-group G which is an E-group is called a “pE-group″. Every abelian group is obviously an E-group. We prove that every 2-generator E-group is abelian and that all 3-generator E-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator E-group is abelian. We conjecture that every finite 3-generator E-group should be abelian. Moreover, we show that the minimum order of a non-abelian pE-group is p ⁸ for any odd prime number p and this order is 2⁷ for p = 2. Some of these results are proved for a class wider than the class of E-groups.     

Torsion-free constructive nilpotent <Emphasis Type="Italic">R</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-groups

We consider a torsion-free nilpotent R _p -group, the p-rank of whose quotient by the commutant is equal to 1 and either the rank of the center by the commutant is infinite or the rank of the group by the commutant is finite. We prove that the group is constructivizable if and only if it is isomorphic to the central extension of some divisible torsion-free constructive abelian group by some torsion-free constructive abelian R _p -group with a computably enumerable basis and a computable system of commutators. We obtain similar criteria for groups of that type as well as divisible groups to be positively defined. We also obtain sufficient conditions for the constructivizability of positively defined groups.     

Residual <Emphasis Type="Italic">p</Emphasis>-finiteness of generalized free products of groups

Let p be a prime number. Recall that a group G is said to be a residually finite p-group if for every non-identity element a of G there exists a homomorphism of the group G onto a finite p-group such that the image of a does not coincide with the identity. We obtain a necessary and sufficient condition for the free product of two residually finite p-groups with finite amalgamated subgroup to be a residually finite p-group. This result is a generalization of Higman’s theorem on the free product of two finite p-groups with amalgamated subgroup.     

A Positivstellensatz for forms on the positive orthant

We shall extend the research on power structure of finite p-groups in Mann (J Algebra 42:121–135, 1976) to locally nilpotentp-groups. Firstly, we obtain that a locally nilpotent \(P_i\)-group G with \(|G:\mho _1(G)|< \infty \) is an extension of a divisible abelian group by a finite p-group. Next we get the structure of infinite locally nilpotent p-groups which are not \(P_i\)-groups, but all of whose proper infinite subgroups are \(P_i\)-groups. Finally, we show that locally nilpotent \(P_i\)-groups with all subgroups subnormal are nilpotent.     

Connected components of the category of elementary abelian p-subgroups

We determine the maximal number of conjugacy classes of maximal elementary abelian subgroups of rank 2 in a finite p-group G, for an odd prime p. Namely, it is p if G has rank at least 3 and it is p+1 if G has rank 2. More precisely, if G has rank 2, there are exactly 1,2,p+1, or possibly 3 classes for some 3-groups of maximal nilpotency class.     

A-Solvability and Mixed Abelian Groups

An abelian group A is an S-group (S ⁺-group) if every subgroup B ≤ A of finite index is A-generated (A-solvable). This article discusses some of the differences between torsion-free S-groups and mixed S-groups, and studies (mixed) S- and S ⁺-groups, which are self-small and have finite torsion-free rank.     

On genus and embeddings of torsion-free nilpotent groups of class two

Summary We study embeddings between torsion-free nilpotent groups having isomorphic localizations. Firstly, we show that for finitely generated torsion-free nilpotent groups of nilpotency class 2, the property of having isomorphicP-localizations (whereP denotes any set of primes) is equivalent to the existence of mutual embeddings of finite index not divisible by any prime inP. We then focus on a certain family Γ of nilpotent groups whose Mislin genera can be identified with quotient sets of ideal class groups in quadratic fields. We show that the multiplication of equivalence classes of groups in Γ induced by the ideal class group structure can be described by means of certain pull-back diagrams reflecting the existence of enough embeddings between members of each Mislin genus. In this sense, the family Γ resembles the family N₀ of infinite, finitely generated nilpotent groups with finite commutator subgroup. We also show that, in further analogy with N₀, two groups in Γ with isomorphic localizations at every prime have isomorphic localizations at every finite set of primes. We supply counterexamples showing that this is not true in general, neither for finitely generated torsion-free nilpotent groups of class 2 nor for torsion-free abelian groups of finite rank. Supported by DGICYT grant PB94-0725 This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag.     

A generalization of 2-Baer groups

A group in which all cyclic subgroups are 2-subnormal is called a 2-Baer group. The topic of this paper are generalized 2-Baer groups, i.e., groups in which the non-2-subnormal cyclic subgroups generate a proper subgroup of the group. If this subgroup is non-trivial, the group is called a generalized T₂-group. In particular, we provide structure results for such groups, investigate their nilpotency class, and construct examples of finite p-groups which are generalized T₂-groups.     

Asymptotic growth of averaged Dehn functions for nilpotent groups

It is proved that in any finite representation of any finitely generated nilpotent group of nilpotency class l ⩾ 1, the averaged Dehn function σ(n) is subasymptotic w.r.t. the function n^l+1. As a consequence it is stated that in every finite representation of a free nilpotent group of nilpotency class l of finite rank r ⩾ 2, the Dehn function σ(n) is Gromov subasymptotic. Supported by RFBR grant No. 04-01-00489. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 60–74, January–February, 2007.     

A special class of nilmanifolds admitting an Anosov diffeomorphism

A nilmanifold admits an Anosov diffeomorphism if and only if its fundamental group (which is finitely generated, torsion-free and nilpotent) supports an automorphism having no eigenvalues of absolute value one. Here we concentrate on nilpotency class 2 and fundamental groups whose commutator subgroup is of maximal torsion-free rank. We prove that the corresponding nilmanifold admits an Anosov diffeomorphism if and only if the torsion-free rank of the abelianization of its fundamental group is greater than or equal to 3.

     

On noninner automorphisms of 2-generator finite p-groups

A longstanding conjecture asserts that every finite nonabelian p-group admits a noninner automorphism of order p. In this paper we give some necessary conditions for a possible counterexample G to this conjecture, in the case when G is a 2-generator finite p-group. Then we show that every 2-generator finite p-group with abelian Frattini subgroup has a noninner automorphism of order p.     

Some 3-Generator, 3-Relation Finite 2-Groups

Using computational methods, we first show that there are exactly eighteen 3-generator 2-groups of order 2¹⁰ with trivial Schur multiplier all having deficiency zero. We next generalize one of the groups obtained to exhibit two infinite classes of 3-generator, 3-relation finite 2-groups of high nilpotency class providing an affirmative answer to a problem posed by Havas et al.     

A finiteness condition on subgroups of large derived length

The authors discuss the class S_d(r) of groups in which every finitely generated subgroup is either at most r-generated or soluble of derived length at most d. Such groups need not be of finite rank or soluble of derived length at most d in general. A structure theorem is obtained for locally finite, and for certain locally nilpotent, S_d(r)-groups.     

On finite alperin <Emphasis Type="Italic">p</Emphasis>-groups with homocyclic commutator subgroup

We study metabelian Alperin groups, i.e., metabelian groups in which every 2-generated subgroup has a cyclic commutator subgroup. It is known that, if the minimum number d(G) of generators of a finite Alperin p-group G is n ≥ 3, then d(G′) ≤ C _n ² for p≠ 3 and d(G′) ≤ C _n ² + C _n ³ for p = 3. The first section of the paper deals with finite Alperin p-groups G with p≠ 3 and d(G) = n ≥ 3 that have a homocyclic commutator subgroup of rank C _n ² . In addition, a corollary is deduced for infinite Alperin p-groups. In the second section, we prove that, if G is a finite Alperin 3-group with homocyclic commutator subgroup G- of rank C _n ² + C _n ³ , then G″ is an elementary abelian group.     

Finite extensions and unipotent shadows of affine crystallographic groupsExtensions finies et ombres unipotentes des groupes affines cristallographiques

Let Γ be a virtually polycyclic group so that the Fitting subgroup is torsion-free and contains its centralizer. We prove that an effective extension of Γ by a finite group μ is isomorphic to an affine crystallographic group if and only if there exists a fixed point for the action of μ on the deformation space of affine crystallographic actions of Γ. We associate to Γ a finitely generated torsion-free nilpotent group Θ which is called the unipotent shadow of Γ, and we relate the deformation space of Γ to the deformation space of Θ. As an application, we show that Γ is isomorphic to an affine crystallographic group if, e.g., Θ has nilpotency class ?3, or if the polycylic rank of Γ is ?5, and also in some other cases. To cite this article: O. Baues, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 785–788.     

On the Approximability by Finite p-Groups of Free Products of Groups with Normal Amalgamation

A sufficient condition for the residual p-finiteness (approximability by the class of finite p-groups) of a free product G = (A * B; H) of groups A and B with a normal amalgamated subgroup H is obtained. This condition is used to prove that if A and B are extensions of residually -groups by -groups, where stands for the class of finitely generated torsion-free nilpotent groups, and if H is a normal p′-isolated polycyclic subgroup, then the group G is residually p-finite (i.e., residually -group), provided the quotient group G/H ^p H′ is residually p-finite.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 125–131.Original Russian Text Copyright © 2005 by E. V. Sokolov.     

Semivarieties of nilpotent groups

Semivarieties of groups are quasivarieties defined by quasi-identities of the form t = 1 → f = 1. It is proved that a set of semivarieties in every variety of class two nilpotent p-groups of finite exponent having a commutator subgroup of exponent p (p is a prime) is at most countable. It is stated that a variety of class two nilpotent groups with commutator subgroup of exponent p contains a set of semivarieties of the cardinality of the continuum.     

Centralizers of coprime automorphisms of finite groups

Let A be an elementary abelian group of order p ^k with k ≥ 3 acting on a finite p′-group G. The following results are proved. If γ _k-2(C _G (a)) is nilpotent of class at most c for any ${a \in A^{\#}}$ , then γ _k-2(G) is nilpotent and has {c, k, p}-bounded nilpotency class. If, for some integer d such that 2 ^d  + 2 ≤ k, the dth derived group of C _G (a) is nilpotent of class at most c for any ${a \in A^{\#}}$ , then the dth derived group G ^(d) is nilpotent and has {c, k, p}-bounded nilpotency class.     

Frattini subgroups ofp-closed andp-supersolvable groups

A nonabelianp-group with cyclic center cannot occur as a normal subgroup contained in the Frattini subgroup of ap-closed group. If a nonabelian normal subgroup of orderp ⁿ and nilpotence classk is contained in the Frattini subgroup of ap-closed group, then its exponent is a divisor ofp ^n−k . This fact is used to derive a relation among the order, number of generators, exponent, and class of the Frattini subgroup, forp-groups. Finally, it is conjectured that a nonabelianp-group having center of orderp cannot occur as a normal subgroup contained in the Frattini subgroup of any finite group. A proof is given forp-supersolvable groups.     

Maximal abelian and minimal nonabelian subgroups of some finite two-generator p-groups especially metacyclic

We study the subgroup structure of some two-generator p-groups and apply the obtained results to metacyclic p-groups. For metacyclic p-groups G, p > 2, we do the following: (a) compute the number of nonabelian subgroups with given derived subgroup, show that (ii) minimal nonabelian subgroups have equal order, (c) maximal abelian subgroups have equal order, (d) every maximal abelian subgroup is contained in a minimal nonabelian subgroup and all maximal subgroups of any minimal nonabelian subgroup are maximal abelian in G. We prove the same results for metacyclic 2-groups (e) with abelian subgroup of index p, (f) without epimorphic image ? D₈. The metacyclic p-groups containing (g) a minimal nonabelian subgroup of order p ⁴, (h) a maximal abelian subgroup of order p ³ are classified. We also classify the metacyclic p-groups, p > 2, all of whose minimal nonabelian subgroups have equal exponent. It appears that, with few exceptions, a metacyclic p-group has a chief series all of whose members are characteristic.