Automorphism Groups of Generalized (Binary) Icosahedral, Tetrahedral and Octahedral Groups

Let G be any of the (binary) icosahedral, generalized octahedral (tetrahedral) groups or their quotients by the center. We calculate the automorphism group Aut(G).

Twisted conjugacy classes in nilpotent groups

A group is said to have the R(infinity) property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether G has the R(infinity) property when G is a finitely generated torsion-free nilpotent group. As a consequence, we show that for every positive integer n >= 5, there is a compact nilmanifold of dimension n on which every homeomorphism is isotopic to a fixed point free homeomorphism. As a by-product, we give a purely group theoretic proof that the free group on two generators has the R(infinity) property. The R(infinity) property for virtually abelian and for C-nilpotent groups are also discussed.

On automorphisms of split metacyclic groups

Let D( m, n; k) be the semi-direct product of two finite cyclic groups Z/m = < x > and Z/n = < y >, where the action is given by yxy(-1) = x(k). In particular, this includes the dihedral groups D(2m). We calculate the automorphism group Aut (D(m, n; k)).

On cohomologies and extensions of cyclic groups

Cohomology groups H(s)(Z(n), Z(m)) are studied to describe all groups up to isomorphism which are (central) extensions of the cyclic group Z(n) by the Z(n)-module Z(m). Further, for each such a group the number of non-equivalent extensions is determined. (C) 2011 Elsevier B.V. All rights reserved.

On automorphisms of finite abelian p-groups

Let A be a finitely generated abelian group. We describe the automorphism group Aut(A) using the rank of A and its torsion part p-part A(p). For a finite abelian p-group A of type (k(1),..., k(n)), simple necessary and sufficient conditions for an n x n-matrix over integers to be associated with an automorphism of A are presented. Then, the automorphism group Aut(A) for a finite p-group A of type (k(1), k(2)) is analyzed. (C) 2008 Mathematical Institute Slovak Academy of Sciences.

CLASSIFICATION OF SUBALGEBRAS OF THE CAYLEY ALGEBRA OVER A FINITE FIELD

We classify all unital subalgebras of the Cayley algebra O(q) over the finite field F(q), q = p(n). We obtain the number of subalgebras of each type and prove that all isomorphic subalgebras are conjugate with respect to the automorphism group of O(q). We also determine the structure of the Moufang loops associated with each subalgebra of O(q).

SIGMA THEORY AND TWISTED CONJUGACY CLASSES

Using Sigma theory we show that for large classes of groups G there is a subgroup H of finite index in Aut(G) such that for phi is an element of H the Reidemeister number R(phi) is infinite. This includes all finitely generated nonpolycyclic groups G that fall into one of the following classes: nilpotent-by-abelian groups of type FP(infinity); groups G/G `` of finite Prufer rank; groups G of type FP(2) without free nonabelian subgroups and with nonpolycyclic maximal metabelian quotient; some direct products of groups; or the pure symmetric automorphism group. Using a different argument we show that the result also holds for 1-ended nonabelian nonsurface limit groups. In some cases, such as with the generalized Thompson`s groups F(n,0) and their finite direct products, H = Aut(G).

Spherical space forms - Homotopy self-equivalences and homotopy types, the case of the groups Z/a x (Z/b x TL(2)(F(p)))

Let G = Z/a x(mu) (Z/b x TL(2)(F(p))) and X(n) be an n-dimensional CW-complex with the homotopy type of the n-sphere. We determine the automorphism group Aut(G) and then compute the number of distinct homotopy types of spherical space forms with respect to free and cellular G-actions on all CW-complexes X(2dn - 1), where 2d is a period of G. Next, the group E(X(2dn - 1)/alpha) of homotopy self-equivalences of spherical space forms X(2dn - 1)/alpha, associated with such G-actions alpha on X(2dn - 1) are studied. Similar results for the rest of finite periodic groups have been obtained recently and they are described in the introduction. (C) 2009 Elsevier B.V. All rights reserved.

TWISTED CONJUGACY CLASSES IN R. THOMPSON`S GROUP F

In this article, we prove that any automorphism of R. Thompson`s group F has infinitely many twisted conjugacy classes. The result follows from the work of Brin, together with standard facts about R. Thompson`s group F, and elementary properties of the Reidemeister numbers.