Mitochondrial DNA Polymorphism of The European Roe Deer, Capreolus Capreolus (Artiodactyla, Cervidae), from The South-West of Ukraine

Analysis of mtDNA control region (934 b. p.) and cytochrome b gene (1140 b. p.) polymorphism of the 33 roe deer samples from the south-west of Ukraine was carried out. 30 different haplotypes of mtDNA have been described and all of them are related to Capreolus capreolus Linnaeus, 1758. Two well differentiated haplogroups were discovered on the examined territory. The isolated subgroup of related haplotypes was revealed in the Crimea that is likely a result of the long-term geographical isolation of the Crimean roe deer populations.

On p-Separability of Subgroups of Free Metabelian Groups

We prove that every free metabelian non-cyclic group has a finitely generated isolated subgroup which is not separable in the class of nilpotent groups. As a corollary, we prove that for every prime number p, an arbitrary free metabelian non-cyclic group has a finitely generated p′-isolated subgroup which is not p-separable.

On the Residual Finiteness of Polygonal Products of Nilpotent Groups

In general polygonal products of finitely generated torsion-free nilpotent groups amalgamating cyclic subgroups need not be residually finite. In this paper we prove that polygonal products of finitely generated torsion-free nilpotent groups amalgamating maximal cyclic subgroups such that the amalgamated cycles generate an isolated subgroup in the vertex group containing them, are residually finite. We also prove that, for finitely generated torsion-free nilpotent groups, if the subgroups generated by the amalgamated cycles have the same nilpotency classes as their respective vertex groups, then their polygonal product is residually finite.

On Torsion-Free Groups of Finite Rank

This paper deals with two conditions which, when stated, appear similar, but when applied to finitely generated solvable groups have very different effect. We first establish the notation before stating these conditions and their implications. If H is a subgroup of a group G, let denote the setWe say G has the isolator property if is a subgroup for all H ≦ G. Groups possessing the isolator property were discussed in [2]. If we define the relation ∼ on the set of subgroups of a given group G by the rule H ∼ K if and only if , then ∼ is an equivalence relation and every equivalence class has a maximal element which may not be unique. If , we call H an isolated subgroup of G.

Cyclically separated groups

We call a group G cyclically separated if for any given cyclic subgroup B in G and subgroup A of finite index in B, there exists a normal subgroup N of G of finite index such that N ∩ B = A. This is equivalent to saying that for each element x ∈ G and integer n ≥ 1 dividing the order o(x) of x, there exists a normal subgroup N of G of finite index such that Nx has order n in G/N. As usual, if x has infinite order then all integers n ≥ 1 are considered to divide o(x). Cyclically separated groups, which are termed “potent groups” by some authors, form a natural subclass of residually finite groups and finite cyclically separated groups also form an interesting class whose structure we are able to describe reasonably well. Construction of finite soluble cyclically separated groups is given explicitly. In the discussion of infinite soluble cyclically separated groups we meet the interesting class of Fitting isolated groups, which is considered in some detail. A soluble group G of finite rank is Fitting isolated if, whenever H = K/L (L ⊲ K ≤ G) is a torsion-free section of G and F(H) is the Fitting subgroup of H then H/F(H) is torsion-free abelian. Every torsion-free soluble group of finite rank contains a Fitting isolated subgroup of finite index.