Unions of dihedral groups

By the following simple formula (1) $$\forall x \exists y (x = xyy, y = xyx)$$ We characterize semigroups from the title. Considering a local property of their ?-classes we get bands and Boolean groups as extreme cases of semigroups with that property. We also provide a construction showing that ?-classes can be sufficiently complicated (at least as Abelian groups are). Then we permute right-hand sides of identities in (1) getting Boolean semigroups (x³=x) and so-called anti-inverse semigroups. Finally we show that Boolean semigroups are a proper subclass of the intersection of anti-inverse semigroups and unions of dihedral groups.     

Limits of dihedral groups

We characterise limits of dihedral groups in the space of finitely generated marked groups. We also describe the topological closure of dihedral groups in the space of marked groups on a fixed number of generators.     

Class groups of dihedral extensions

Let L/F be a dihedral extension of degree 2p, where p is an odd prime. Let K/F and k/F be subextensions of L/F with degrees p and 2, respectively. Then we will study relations between the p‐ranks of the class groups Cl(K) and Cl(k). (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finitary automorphisms of groups

We introduce the so-called multiscale limit for spectral curves associated with real finite-gap sine-Gordon solutions. This technique allows us to solve the old problem of calculating the density of the topological charge for real finite-gap sine-Gordon solutions directly from the θ-functional formulae.     

Infinite locally dihedral groups as automorphism groups

If \(A\) is a nontrivial torsion-free, locally cyclic group with no nontrivial divisible quotients, and \(G\) is the split extension of \(A\) by a group of order 2 acting on \(A\) by means of the inverting map, then \(G\simeq {{{\mathrm{Aut}}}G} \). We prove that in no other case the full automorphism group of a group is infinite and locally dihedral.     

Tate cohomologies and Browder-Livesay groups of dihedral groups

Translated from Matematicheskie Zametki, Vol. 54, No. 2, pp. 44–55, August, 1993.     

Difference sets in dihedral groups

We study the existence of nontrivial (2m, k, )-difference sets in dihedral groups. Some nonexistence results are proved. In particular, we show that n = k – is odd and (n)/n < 1/2. Finally, a computer search shows that, except 5 undecided cases, no nontrivial difference set exists in dihedral groups for n 10⁶.     

On inner automorphisms and certain central automorphisms of groups

Let G be a group, let M and N be two normal subgroups of G. We denote by Aut _N ^M (G), the set of all automorphisms of G which centralize G/M and N. In this paper we investigate the structure of a group G in which one of the Inn(G) = Aut _N ^M (G), Aut _N ^M (G) ≤ Inn(G) or Inn(G) ≤ Aut _N ^M (G) holds. We also discuss the problem: “what conditions on G is sufficient to ensure that G has a non-inner automorphism which centralizes G/M and N”.     

Affine buildings for dihedral groups

We construct 2-dimensonal thick nondiscrete affine buildings associated with an arbitrary finite dihedral group.     

Blocks with dihedral defect groups in solvable groups

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