Meta-Hamiltonian groups with elementary commutant of rank 2

This paper is concerned with the structure of periodic meta-Hamiltonian groups with elementary commutant of rank 2 that have complemented Miller Moreno subgroups. It is proved that there exist four types of such groups.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 168–175, February, 1990.The proof of this assertion will be given in another paper of the authors.     

On the projective commutant of abelian groups

We describe the projective commutant and projective center of nonreduced groups and also of separable, vector, algebraically compact, and some other abelian groups.     

A characterization of elementary abelian 2-groups

In this note we give a characterization of elementary abelian 2-groups in terms of their maximal sum-free subsets.     

On finite Alperin 2-groups with elementary abelian second commutants

By an Alperin group we mean a group in which the commutant of each 2-generated subgroup is cyclic. Alperin proved that if p is an odd prime then all finite p-groups with this property are metabelian. The today??s actual problem is the construction of examples of nonmetabelian finite Alperin 2-groups. Note that the author had given some examples of finite Alperin 2-groups with second commutants isomorphic to Z ₂ and Z ₄ and proved the existence of finite Alperin 2-groups with cyclic second commutants of however large order by appropriate examples. In this article the existence is proved of finite Alperin 2-groups with abelian second commutants of however large rank.     

Interlacing of elementary abelian groups

A lattice of characteristic subgroups of multiple interlacings of finite elementary abelian groups by itself is established herein.Translated from Matematicheskie Zametki, Vol. 11, No. 1, pp. 61–72, January, 1972.The author takes this opportunity to thank L. A. Kaloujnine for formulating the problem and for aid in performing the research.     

Endomorphisms of abelian varieties and points of finite order in characteristic p

An analog of the Tate hypothesis on homomorphisms of Abelian varieties is proved, in which points of sufficiently large prime order figure in place of the Tate modules. As is the case with the Tate hypothesis, this assertion follows formally from a finiteness hypothesis for isogenies of Abelian varieties, which is proved in characteristic p > 2 and for finite fields. The same methods are used to prove the finiteness of the set of Abelian varieties of a given dimension over a finite field.Translated from Matematicheskie Zametki, Vol. 21, No. 6, pp. 737–744, June, 1977.     

On the construction of CDN[]-groups with elementary commutant of rank two

We describe certain CDN-groups of order p ⁿ with elementary commutant of rank two. Scientific-Research Pedagogic Institute, Ukrainian Pedagogic Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 10, pp. 1396–1403, October, 1997     

Investigations on association schemes with elementary abelian thin residue

In this paper, we give a new class of association schemes whose thin residues are isomorphic to an elementary abelian $p$-group of order $p^2$. We then study the automorphism groups of these schemes and determine whether these schemes are schurian.     

Structure of a group with elements of order at most 4

We prove that every group in which the order of each element is at most 4 either possesses a nontrivial class 2 nilpotent normal Sylow subgroup or includes a normal elementary abelian 2-subgroup the quotient by which is isomorphic to the nonabelian group of order 6.     

Flag-Transitive affine planes of order at most 125

We complete the determination of all the flag-transitive affine planes of order at most 125. In particular, all such planes of order 81 are determined. We show that those with a solvable collineation group are obtained by constructions due to Kantor and Suetake.