Finite quaternion-free 2-groups

We present an elementary proof of the classification theorem for finite nonmodular quaternion-free 2-groups. This proof does not involve the structure theory of powerful 2-groups. Such a new proof is also necessary, since there are several gaps in the original proof given in [5].     

Finite separable metacyclic 2-groups

A finite separable metacyclic 2-group G can be written as the semidirect product of a cyclic group with another cyclic group. Necessary and sufficient conditions are given for when all other split decompositions of G result in, up to isomorphism, this same semidirect product representation for G.     

Soluble (HN)2-groups

In this paper we investigate the class of finite soluble groups in which every subnormal subgroup has normal normalizer. In particular we prove that they areUN ₂U, whereU andN ₂denote finite abelian groups and of finite nilpotent groups of class at most 2 respectively.     

Structure ofR(3,3)-groups

We describe groupsG in which the set {abc, acb, bac, bca, cab, cba} contains three different elements at most for anya, b, c ∈G and show how this type of problem is connected with the rewritability (or permutation) properties of groups.     

Finite 2-groups having invariant non-completely partitionable subgroups

A theorem is proved which classifies and describes up to generating elements and defining relations the finite 2-groups having invariant non-completely partitionable subgroups.Translated from Matematicheskie Zametki, Vol. 10, No. 2, pp. 151–161, August, 1971.The author is grateful to N. F. Sesekin for the problem statement and scientific direction.     

Soluble minimal non-(finite-by-Baer)-groups

Let \mathfrakX{\mathfrak{X}} be a class of groups. A group G is called a minimal non- \mathfrakX{\mathfrak{X}}-group if it is not an \mathfrakX{\mathfrak{X}}-group but all of whose proper subgroups are \mathfrakX{\mathfrak{X}}-groups. In [16], Xu proved that if G is a soluble minimal non-Baer-group, then G/G ^′′ is a minimal non-nilpotent-group which possesses a maximal subgroup. In the present note, we prove that if G is a soluble minimal non-(finite-by-Baer)-group, then for all integer n ≥ 2, G/γ _n (G′) is a minimal non-(finite-by-abelian)-group.     

Finite alperin 2-groups with cyclic second commutants

An Alperin group is a group in which every 2-generated subgroup has a cyclic commutant. Previously, we constructed examples of finite Alperin 2-groups with second commutant isomorphic to Z ₂ or Z ₄. Here, it is proved that for any natural n, there exists a finite Alperin 2-group whose second commutant is isomorphic to Z _2n .     

Finite 2-groups with exactly one nonmetacyclic maximal subgroup

We determine here the structure of the title groups. All such groups G will be given in terms of generators and relations, and many important subgroups of these groups will be described. Let d(G) be the minimal number of generators of G. We have here d(G) ≤ 3 and if d(G) = 3, then G′ is elementary abelian of order at most 4. Suppose d(G) = 2. Then G′ is abelian of rank ≤ 2 and G/G′ is abelian of type (2, 2^m), m ≥ 2. If G′ has no cyclic subgroup of index 2, then m = 2. If G′ is noncyclic and G/Φ(G ₀) has no normal elementary abelian subgroup of order 8, then G′ has a cyclic subgroup of index 2 and m = 2. But the most important result is that for all such groups (with d(G) = 2) we have G = AB, for suitable cyclic subgroups A and B. Conversely, if G = AB is a finite nonmetacyclic 2-group, where A and B are cyclic, then G has exactly one nonmetacyclic maximal subgroup. Hence, in this paper the nonmetacyclic 2-groups which are products of two cyclic subgroups are completely determined. This solves a long-standing problem studied from 1953 to 1956 by B. Huppert, N. Itô and A. Ohara. Note that if G = AB is a finite p-group, p > 2, where A and B are cyclic, then G is necessarily metacyclic (Huppert [4]). Hence, we have solved here problem Nr. 776 from Berkovich [1].     

Finite Gröbner basis algebra with unsolvable problem of zero divisors

This work presents a sample construction of an algebra with the ideal of relations defined by a finite Gröbner basis for which the question whether this element is a zero divisor is algorithmically unsolvable. This gives the negative answer to a question raised by V. N. Latyshev.     

所有子群皆循环或正规的有限2-群

研究了所有子群皆循环或正规的有限2群,并给出了其完全分类.     

A note on unsolvable systems of max–min (fuzzy) equations

For unsolvable systems of linear equations of the form Ax=b over the max–min (fuzzy) algebra

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we propose an efficient method for finding a Chebychev-best approximation of the matrix in the set

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.     

A condition of indecomposability for Butler \mathrm B (n)-groups

We give a manageable sufficient condition for indecomposability of Butler \(\mathrm B (n)\) -groups, allowing the easy construction of a big family of indecomposable torsionfree Abelian groups of finite rank.