Residual finiteness, subgroup separability and conjugacy separability of certain HNN extensions

In this note we shall give characterisations for HNN extensions of non-cyclic polycyclic-by-finite groups with normal infinite cyclic associated subgroups to be residually finite, subgroup separable and conjugacy separable.     

Virtual retractions,conjugacy separability and omnipotence

We use wreath products to provide criteria for a group to be conjugacy separable or omnipotent. These criteria are in terms of virtual retractions onto cyclic subgroups. We give two applications: a straightforward topological proof of the theorem of Stebe that infinite-order elements of Fuchsian groups (of the first type) are conjugacy distinguished, and a proof that surface groups are omnipotent.     

Conjugacy separability of 1-acylindrical graphs of free groups

We use the theory of group actions on profinite trees to prove that the fundamental group of a finite, 1-acylindrical graph of free groups with finitely generated edge groups is conjugacy separable. This has several applications: we prove that positive, C′(1/6) one-relator groups are conjugacy separable; we provide a conjugacy separable version of the Rips construction; we use this latter to provide an example of two finitely presented, residually finite groups that have isomorphic profinite completions, such that one is conjugacy separable and the other does not even have solvable conjugacy problem.     

Bianchi groups are conjugacy separable

We prove that non-uniform arithmetic lattices of SL₂(C) and consequently the Bianchi groups are conjugacy separable. The proof is based on recent deep results of Agol, Long, Reid and Minasyan. The conjugacy separability of groups commensurable with limit groups is also established.     

Conjugacy separability of Seifert 3-manifold groups over non-orientable surfaces

Scott (1978) [12] showed Seifert 3-manifold groups are subgroup separable. Niblo (1992) [9] improved this result by showing that these groups are double coset separable. In Allenby, Kim and Tang (2005) [2] it was shown that all but two types of groups in the orientable case are conjugacy separable. Martino (2007) [7] using topological results showed that Seifert groups are conjugacy separable. Here we use algebraic method to show that Seifert groups over non-orientable surfaces are conjugacy separable.     

Finite groups have even more conjugacy classes

In his paper Finite groups have many conjugacy classes (J. London Math. Soc (2) 46 (1992), 239–249), L. Pyber proved the to-date best general lower bounds for the number of conjugacy classes of a finite group in terms of the order of the group. In this paper we strengthen the main results in Pyber’s paper.     

On the number of conjugacy classes of zeros of characters and the length of solvable groups

Abstract

We apply an orbit theorem to a few questions about character degrees. We investigate the relation of the number of conjugacy classes where characters vanish and the length of the solvable groups. As another application, we give a bound for the size of defect groups of blocks of solvable groups.

Communicated by J. Zhang     

Quasiconcavity of a separable product of utility functions

ABSTRACT

The nontrivial problem of the quasiconcavity of a separable sum of utility functions has been studied and solved by Debreu-Koopmans and Crouzeix-Lindberg via the introduction of convexity indices. This paper studies the equivalent problem of the quasiconcavity of a separable products. This approach makes the proofs easier.     

On stable conjugacy of finite subgroups of the plane Cremona group, I

We discuss the problem of stable conjugacy of finite subgroups of Cremona groups. We compute the stable birational invariant H ¹(G, Pic(X)) for cyclic groups of prime order.     

Theorem about the conjugacy representation ofS n

Every group has two natural representations on itself, the regular representation and the conjugacy representation. We know everything about the construction of the regular representation, but we know very little about the conjugacy representation (for uncommutative groups). In this paper we will see that every irreducible complex character ofS _n (n>2) is a constituent of conjugacy character ofS _n .     

Sign conjugacy classes of the alternating groups

A conjugacy class C of a finite group G is a sign conjugacy class if every irreducible character of G takes value 0,1 or ?1 on C. In this paper, we classify the sign conjugacy classes of alternating groups.     

A note on conjugacy classes of finite groups

Let G be a finite group and let x ^G denote the conjugacy class of an element x of G. We classify all finite groups G in the following three cases: (i) Each non-trivial conjugacy class of G together with the identity element 1 is a subgroup of G, (ii) union of any two distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G, and (iii) union of any three distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G.     

Using Mathematica to compute conjugacy classes

The conjugacy classes of finite groups play an important role in the representation theory of those groups, and it is useful to be able to compute the conjugacy classes quickly. A procedure is developed and then implemented with Mathematica to discover these conjugacy classes. The computations make use of the Cayley table in its regular form for the group. The conjugacy classes for C_4v, the point symmetry group of the square, are displayed.     

A remark on local-global principles for conjugacy classes

A local-global principle is shown to hold for all conjugacy classes of any inner form of GL(n), SL(n), U(n), SU(n), and for all semisimple conjugacy classes in any inner form of Sp(n), over fieldsk with vcd(k)≤1. Over number fields such a principle is known to hold for any inner form of GL(n) and U(n), and for the split forms of Sp(n), O(n), as well as for SL(p) but not for SL(n),n non-prime. The principle holds for all conjugacy classes in any inner form of GL(n), but not even for semisimple conjugacy classes in Sp(n), over fieldsk with vcd(k)≤2. The principle for conjugacy classes is related to that for centralizers.     

Non-monotonic solutions and continuously differentiable solutions of conjugacy equations

In this paper solutions of conjugacy equation φ(f(x))=g(φ(x)) for a strictly decreasing continuous given function f and a continuous given function g (maybe non-monotonic) are constructed by piecewise defining. We determine the conditions for piecewise continuously differentiable solutions of conjugacy equations with a strictly decreasing continuously differentiable given function f and a continuously differentiable given function g. Finally, the recursive algorithm is implemented in MATLAB software and two examples are respectively presented for a non-monotonic solution and a continuously differentiable one.     

Derived length and products of conjugacy classes

Let G be a supersolvable group and A be a conjugacy class of G. Observe that for some integer η(AA ⁻¹) > 0, AA ⁻¹ = {ab ⁻¹: a, b ∈ A} is the union of η(AA ⁻¹) distinct conjugacy classes of G. Set C _G (A) = {g ∈ G: a ^g = a for all a ∈ A. Then the derived length of G/C _G (A) is less or equal than 2η(AA ⁻¹) − 1.     

The conjugacy problem for Dehn twist automorphisms of free groups

A Dehn twist automorphism of a group G is an automorphism which can be given (as specified below) in terms of a graph-of-groups decomposition of G with infinite cyclic edge groups. The classic example is that of an automorphism of the fundamental group of a surface which is induced by a Dehn twist homeomorphism of the surface. For , a non-abelian free group of finite rank n, a normal form for Dehn twist is developed, and it is shown that this can be used to solve the conjugacy problem for Dehn twist automorphisms of . Received: February 12, 1996.     

(Cyclic) subgroup separability of HNN and split extensions

This work has been divided in two parts. In the first part we give a sufficient conditions on an HNN extension of a free group to be cyclic subgroup seperable. In the second part we define just subgroup separability on a split extension of special groups which is actually on holomorph.