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.     

Projective Character Degree Patterns of p-Groups for p Odd

In this article, we classify up to isomorphism the character tables of 2-generator p-groups of class two.     

Subgroups of powerful groups

We consider subgroups of powerfulp-groups. In particular, we give a new proof that allp-groups are sections of powerfulp-groups, give necessary and sufficient conditions for a 2-generator group to be a normal subgroup of a powerfulp-groups, and show thatp-groups of class 2, orp-groups with a cyclic commutator subgroup, are such normal subgroups.     

Counterexamples to a rank analog of the Shepherd-Leedham-Green-Mckay theorem on finite p-groups of maximal nilpotency class

By the Shepherd-Leedham-Green-McKay theorem on finite p-groups of maximal nilpotency class, if a finite p-group of order p ⁿ has nilpotency class n?1, then f has a subgroup of nilpotency class at most 2 with index bounded in terms of p. Some counterexamples to a rank analog of this theorem are constructed that give a negative solution to Problem 16.103 in The Kourovka Notebook. Moreover, it is shown that there are no functions r(p) and l(p) such that any finite 2-generator p-group whose all factors of the lower central series, starting from the second, are cyclic would necessarily have a normal subgroup of derived length at most l(p) with quotient of rank at most r(p). The required examples of finite p-groups are constructed as quotients of torsion-free nilpotent groups which are abstract 2-generator subgroups of torsion-free divisible nilpotent groups that are in the Mal’cev correspondence with “truncated” Witt algebras.     

Capable 2-Generator 2-Groups of Class Two

A group is called capable if it is a central factor group. We characterize the capable 2-generator 2-groups of class 2 in terms of a standard presentation.     

The cohomology of homotopy categories and the general linear group

A homotopy categoryC (of co-H-groups resp.H-groups) represents an element C in the third cohomology ofC. This element determines all Toda brackets and secondary homotopy operations inC. Moreover, in caseC =VS ⁿ consists of all one-point unions ofn-spheres, the bracket is actually a /2-generator which restricts to Igusa's class(1) in casen3; an explicit new cocycle for(1) is obtained by automorphisms of free nil(2)-groups.     

Semistrict models of connected 3-types and Tamsamani’s weak 3-groupoids

Homotopy 3-types can be modelled algebraically by Tamsamani’s weak 3-groupoids as well as, in the path-connected case, by cat²-groups. This paper gives a comparison between the two models in the path-connected case. This leads to two different semistrict algebraic models of connected 3-types using Tamsamani’s model. Both are then related to Gray groupoids.     

New negative Latin square type partial difference sets in nonelementary abelian 2-groups and 3-groups

A partial difference set having parameters (n ², r(n − 1), n + r ² − 3r, r ² − r) is called a Latin square type partial difference set, while a partial difference set having parameters (n ², r(n + 1), − n + r ² + 3r, r ² + r) is called a negative Latin square type partial difference set. Nearly all known constructions of negative Latin square partial difference sets are in elementary abelian groups. In this paper, we develop three product theorems that construct negative Latin square type partial difference sets and Latin square type partial difference sets in direct products of abelian groups G and G′ when these groups have certain Latin square or negative Latin square type partial difference sets. Using these product theorems, we can construct negative Latin square type partial difference sets in groups of the form where the s _i are nonnegative integers and s ₀ + s ₁ ≥ 1. Another significant corollary to these theorems are constructions of two infinite families of negative Latin square type partial difference sets in 3-groups of the form for nonnegative integers s _i . Several constructions of Latin square type PDSs are also given in p-groups for all primes p. We will then briefly indicate how some of these results relate to amorphic association schemes. In particular, we construct amorphic association schemes with 4 classes using negative Latin square type graphs in many nonelementary abelian 2-groups; we also use negative Latin square type graphs whose underlying sets can be elementary abelian 3-groups or nonelementary abelian 3-groups to form 3-class amorphic association schemes.      

Finite 2-generator equilibrated p-groups

Following Blackburn, Deaconescu and Mann, a group G is called an equilibrated group if for any subgroups H,K of G with HK = KH, either H≤NG(K) or K≤NG(H). Continuing their work and based on the classification of metacyclic p-groups given by Newman and Xu, we give a complete classification of 2-generator equilibrated p-groups in this note.     

On the Derived Series of Coxeter Groups

Let G be a group and let φ(G) be the least integer k such that G^(k) = G^(k+1). If no such k exists, then φ(G) = ∞ and we write G ∈ 𝒰. We are interested in the questions which Coxeter groups are in 𝒰 and how large can finite φ(G) be for Coxeter groups. The second author answered these questions for 3-generator and 4-generator Coxeter groups. This article begins the study for the 5-generator case.     

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.     

Finite p-groups with exactly one A 1-subgroup of given structure of order p 3

Abstract In this paper, we classified the finite p-groups with exactly one A1-subgroup of given structure of order p^3.     

Finite p-groups whose nonnormal subgroups have orders at most p 3

We classify finite p-groups all of whose nonnormal subgroups have orders at most p3, p odd prime. Together with a known result, we completely solved Problem 2279 proposed by Y. Berkovich and Z. Janko in Groups of Prime Power Order, Vol. 3.     

The K-Theory of Heegaard-Type Quantum 3-Spheres

We use a Heegaard splitting of the topological 3-sphere as a guiding principle to construct a family of its noncommutative deformations. The main technical point is an identification of the universal C^*-algebras defining our quantum 3-spheres with an appropriate fiber product of crossed-product C^*-algebras. Then we employ this result to show that the K-groups of our family of noncommutative 3-spheres coincide with their classical counterparts. Dedicated to the memory of Olaf Richter An erratum to this article is available at .     

Central factors of deficiency zero groups

We answer a question of some twenty years standing: are the central factors of nilpotent groups of deficiency zero 3-generated? We prove that the answer is negative by giving an explicit presentation for a 3-generator, 3-relator group of order 2¹⁷ and class 5 which has central factors wl ch are 4-generated but not 3-generated. We outline the computatio al techniques which lead to this result.     

On the Existence of 3-Generator, 3-Relation Finite 2-Groups with a Given (Co)Class

In this article, we show that for any positive integer k there is a 3-generator, 3-relation finite 2-group of class (respectively, coclass) k provided that k ≥ 4 (respectively, k ≥ 5).     

A note on the security of MST 3

In this paper, we study the recently proposed encryption scheme MST ₃, focusing on a concrete instantiation using Suzuki-2-groups. In a passive scenario, we argue that the one wayness of this scheme may not, as claimed, be proven without the assumption that factoring group elements with respect to random covers for a subset of the group is hard. As a result, we conclude that for the proposed Suzuki 2-groups instantiation, impractical key sizes should be used in order to prevent more or less straightforward factorization attacks.     

Finite 2-groups whose nonnormal subgroups have orders at most 23

In this paper, we classify finite 2-groups all of whose nonnormal subgroups have orders at most 2³. Together with a known result, we completely solved Problem 2279 proposed by Y. Berkovich and Z. Janko in Groups of Prime Power Order, Vol. 3.     

Multiplicative Jordan Decomposition in Group Rings of 3-Groups,II

In this paper, we complete the classification of those finite 3-groups G whose integral group rings have the multiplicative Jordan decomposition property. If G is abelian, then it is clear that ?[G] satisfies the multiplicative Jordan decomposition (MJD). In the nonabelian case, we show that ?[G] satisfies MJD if and only if G is one of the two nonabelian groups of order 3³ = 27.     

Infinite p-groups containing exactly p2 solutions of the equation xP=1

We study arbitrary infinite 2-groups with three involutions and infinite locally finite p-groups (p2), containing p²–1 elements of order p. For odd p the groupG=a, where A is a direct product of two quasicyclic 3-groups ¦b¦=9, b³A, and subgroup A is generated by the elements of the commutator ladder of element b, is a unique infinite non-Abelian locally finite p-group whose equation xP=1 has p² solutions.Translated from Matematicheskie Zametki, Vol. 20, No. 1, pp. 11–18, July, 1976.