Coarse structures on groups defined by conjugations

For a group G, we denote by G↔ the coarse space on G endowed with the coarse structure with the base {{(x,y)∈G×G:y∈xF}:F∈[G]<ω}, xF={z−1xz:z∈F}. Our goal is to explore interplays between algebraic properties of G and asymptotic properties of G↔. In particular, we show that asdim G↔=0 if and only if G/ZG is locally finite, ZG is the center of G. For an infinite group G, the coarse space of subgroups of G is discrete if and only if G is a Dedekind group.

Finite groups with metacyclic QTI-subgroups

Let G be a finite group. A subgroup A of G is called a TI-subgroup of G if A ∩ Ax = 1 or A for all x ∈ G. A subgroup H of G is called a QTI-subgroup if CG(x) ⊆ NG(H) for every 1 ≠ x ∈ H, and a group G is called an MCTI-group if all its metacyclic subgroups are QTI-subgroups. In this paper, we show that every nilpotent MCTI-group is either a Dedekind group or a p-group and we completely classify all the MCTI-p-groups. We show that all MCTI-groups are solvable and that every nonnilpotent MCTI-group must be a Frobenius group having abelian kernel and cyclic complement.

Class preserving automorphisms of Blackburn groups

In this article, a Blackburn group refers to a finite non-Dedekind group for which the intersection of all nonnormal subgroups is not the trivial subgroup. By completing the arguments of M. Hertweck, we show that all conjugacy class preserving automorphisms of Blackburn groups are inner automorphisms.

Subgroups like Wielandt's in finite soluble groups

In 1935 Baer[1] introduced the concept of kern of a group as the subgroup of elements normalizing every subgroup of the group. It is of interest from three points of view: that of its structure, the nature of its embedding in the group, and the influence of its internal structure on that of the whole group. The kern is a Dedekind group because all its subgroups are normal. Its structure is therefore known exactly (Dedekind [7]): if not abelian it is a direct product of a copy of the quaternion group of order 8 and an abelian periodic group with no elements of order 4. As for the embedding of the kern, Schenkman[13] shows that it is always in the second centre of the group: see also Cooper [5], theorem 6·5·1. As an example of the influence of the structure of the kern on its parent group we cite Baer's result from [2], p. 246: among 2-groups, only Hamiltonian groups (i.e. non-abelian Dedekind groups) have nonabelian kern.