Conjugacy Classes Of Elements Research Articles

Virtual braid groups, virtual twin groups and crystallographic groups

Let n≥2. Let VBn (resp. VPn) be the virtual braid group (resp. the pure virtual braid group), and let VTn (resp. PVTn) be the virtual twin group (resp. the pure virtual twin group). Let Π be one of the following quotients: VBn/Γ2(VPn) or VTn/Γ2(PVTn) where Γ2(H) is the commutator subgroup of H. In this paper, we show that Π is a crystallographic group and we characterize the elements of finite order and the conjugacy classes of elements in Π. Furthermore, we realize explicitly some Bieberbach groups and infinite virtually cyclic groups in Π. Finally, we also study other braid-like groups (welded, unrestricted, flat virtual, flat welded and Gauss virtual braid group) modulo the respective commutator subgroup in each case.

Designs and codes from fixed points of elements of Janko simple group J 1

We previously have developed two methods for constructing codes and designs from finite groups (mostly simple finite groups). In a recent paper we introduced a new method (Method 3) for constructing codes and designs from fixed points of elements of finite transitive groups. This new method together with background material and results required from finite groups, permutation groups and representation theory were discussed fully in that paper. The main aim of this paper is to apply the method to the first Janko sporadic simple group for the first 3 conjugacy classes of maximal subgroups and its 2A, 3A, 5A and 7A conjugacy classes of elements.

On the Ranks of the Alternating Group $$A_{n}$$ A n

Let G be a finite group and X be a conjugacy class of G. The rank of X in G, denoted by rank(G : X), is defined to be the minimal number of elements of X generating G. In this paper we establish some general results on the ranks of certain conjugacy classes of elements for simple alternating group $$A_{n}$$ . We apply these general results together with the structure constants method to determine the ranks of all the non-trivial classes of $$A_{8}$$ and $$A_{9}$$ .

Conjugation of elements in central simple algebras

ABSTRACTWe study the conjugacy classes of elements of a finite-dimensional central simple associative algebra with respect to scalar extensions. We apply the results to investigate the products of two (skew-)symmetric elements in central simple algebras with involution.

On conjugacy by regularly varying functions

Two selfmappings and of the open interval are said to be regularly conjugate with respect to the distinguished end-point whenever for some uniformly regularly varying at bijective selfmapping we have . Let denote the (Szekeres–Lundberg) family of fixed point free bijections of , strongly attracting to , with derivative at satisfying , which additionally possess continuous (weakly) increasing principal function , for . All regular conjugacy classes of elements of are constructed in terms of the principal functions, the non-measurable included. A few representatives of different regular conjugacy classes among S–L diffeomorphisms are constructed, too.

Groups with finitary classes of conjugate elements

A class of conjugate elements of a group is called finitary if the conjugation action of the group induces a group of finitary permutations of this class. A group with finitary classes of conjugate elements will be called a ΦC-group. Some characterizations of ΦC-groups in the class of all groups have been obtained. It is also shown for every ΦC-group that either it is an FC-group, i.e., a group with finite classes of conjugate elements, or its structure is close to the structure of a totally imprimitive group of finitary permutations.

On Ditsman’s lemma

Let H be a subgroup of a group G generated by a finite G-invariant subset X = tU i=1 k C i that consists of elements of finite order, where C i is a class of conjugate elements of G with representative a i . We prove that $$|H| \leqslant \prod\limits_{i = 1}^k {o(a_i )^{|C_i |} } ,$$ where o(a i ) is the order of the element a i ∈ C i . Best estimates are obtained for some important special cases.

On the ranks of Fi22

If G is a finite group and X a conjugacy class of elements of G, then we define rank(G:X) to be the minimum number of elements of X generating G. Moori investigated rank(Fi22:X), where X is a conjugacy class of involutions in the Fischer group Fi22. In the present article, we completely determine the ranks of the Fischer group Fi22 by computing rank(Fi22:X), where X is any conjugacy class of Fi22.

Groups with Finiteness Conditions on Commutators

The structure of groups for which certain sets of commutator subgroups are finite is investigated. In particular, the relation between such groups and groups with finite conjugacy classes of elements is discussed.

Groups with finitely many conjugacy classes and their automorphisms

We combine classical methods of combinatorial group theory with the theory of small cancellation over relatively hyperbolic groups to construct finitely generated torsion-free groups that have only finitely many classes of conjugate elements. Moreover, we present several results concerning embeddings into such groups. As another application of these techniques, we prove that every countable group C can be realized as a group of outer automorphisms of a group N, where N is a finitely generated group having Kazhdan’s property (T) and containing exactly two conjugacy classes.

On the Ranks of Janko Groups J 1, J 2, J 3 and J 4

If G is a finite group and X a conjugacy class of elements of G, then we define rank(G : X) to be the minimum number of elements of X generating G. In the present paper we study the rank(Ji : X) for all conjugacy classes of Ji , where i = 1, 2, 3, 4.

On a problem from the Kourovka Notebook

In this article, it is proved that if a group G coincides with its commutator subgroup, is generated by a finite set of classes of conjugate elements, and contains a proper minimal normal subgroup A such that the factor group G/A coincides with the normal closure of one element, then G coincides with the normal closure of an element. From this a positive answer to question 5.52 from the Kourovka Notebook for the group with the condition of minimality on normal subgroups follows. We have found a necessary and sufficient condition for a group coinciding with its commutator subgroup and generated by a finite set of classes of conjugate elements not to coincide with the normal closure of any element.

Conjugately dense subgroups of free products of groups with amalgamation

A subgroup having non-empty intersection with each class of conjugate elements of the group is said to be conjugately dense. It is shown that, under certain conditions, the number of conjugately dense subgroups in a free product with amalgamation is not less than some cardinal. As a consequence, P. Neumann’s conjecture in the Kourovka notebook (Question 6.38) is refuted. It is also stated that a modular group and a non-Abelian group of countable or finite rank possess continuum many pairwise non-conjugate conjugately dense subgroups.

Garside groups are strongly translation discrete

In this paper we show that all Garside groups are strongly translation discrete, that is, the translation numbers of non-torsion elements are strictly positive and for any real number r there are only finitely many conjugacy classes of elements whose translation numbers are less than or equal to r. It is a consequence of the inequality “ inf s ( g ) ⩽ inf s ( g n ) n < inf s ( g ) + 1 ” for a positive integer n and an element g of a Garside group G, where inf s denotes the maximal infimum for the conjugacy class. We prove the inequality by studying the semidirect product G ( n ) = Z ⋉ G n of the infinite cyclic group Z and the cartesian product G n of a Garside group G, which turns out to be a Garside group. We also show that the root problem in a Garside group G can be reduced to a conjugacy problem in G ( n ) , hence the root problem is solvable for Garside groups.

CONJUGATELY DENSE SUBGROUPS OF 3-DIMENSIONAL LINEAR GROUPS OVER LOCALLY FINITE FIELD

A subgroup of any group is called conjugately dense if it has nonempty intersection with each class of conjugate elements of the group. The aim of this paper is to prove the following. Let K be a locally finite field and H be an irreducible conjugately dense subgroup of the intermediate group SL 3(K) ≤ G ≤ GL 3(K); then H = G. This result confirms part of P. Neumann's conjecture from problem 6.38 in "Kourovka Notebook" for the group GL 3(K) over locally finite field K.

Order of elements in the groups related to the general linear group

For a natural number n and a prime power q the general, special, projective general and projective special linear groups are denoted by GL n ( q ) , SL n ( q ) , PGL n ( q ) and PSL n ( q ) , respectively. Using conjugacy classes of elements in GL n ( q ) in terms of irreducible polynomials over the finite field GF ( q ) we demonstrate how the set of order elements in GL n ( q ) can be obtained. This will help to find the order of elements in the groups SL n ( q ) , PGL n ( q ) and PSL n ( q ) . We also show an upper bound for the order of elements in SL n ( q ) .

Zeros in tables of characters for the groups Sn and An

In the representation theory of symmetric groups, for each partition α of a natural number n, the partition h(α) of n is defined so as to obtain a certain set of zeros in the table of characters for Sn. Namely, h(α) is the greatest (under the lexicographic ordering ≤) partition among β ∈ P(n) such that χα(gβ) ≠ 0. Here, χα is an irreducible character of Sn, indexed by a partition α, and gβ is a conjugacy class of elements in Sn, indexed by a partition β. We point out an extra set of zeros in the table that we are dealing with. For every non self-associated partition α ∈ P(n), the partition f(α) of n is defined so that f(α) is greatest among the partitions β of n which are opposite in sign to h(α) and are such that χα(gβ) ≠ 0 (Thm. 1). Also, for any self-associated partition α of n > 1, we construct a partition $$\tilde f$$ (α) ∈ P(n) such that $$\tilde f$$ (α) is greatest among the partitions β of n which are distinct from h(α) and are such that χα(gβ) ≠ 0 (Thm. 2).

Groups with Hypercyclic Proper Quotient Groups

We continue the investigation of (solvable) groups all proper subgroups of which are hypercyclic. The monolithic case is studied completely; in the nonmonolithic case, however, one should impose certain additional conditions. We investigate groups all proper quotient groups of which possess supersolvable classes of conjugate elements.

Conjugately Dense Subgroups of Locally Finite Chevalley Groups of Lie Rank 1

Of interest are the subgroups of various groups which have nonempty intersection with each class of conjugate elements of the group under study. We call these subgroups conjugately dense and study Neumann's problem of describing them in the Chevalley groups over a field. The main theorem lists all conjugately dense subgroups of the Chevalley groups of Lie rank 1 over a locally finite field.

Two-transitive actions on conjugacy classes

Every group acts transitively by conjugation on each of its conjugacy classes of elements. It is natural to wonder when this action becomes multiply transitive. In this paper, we determine all finite groups which act faithfully and 2-transitively on a conjugacy class of elements. We also give some consequences including a solvability criterion based on what fraction of elements belong to conjugacy classes upon which the group acts faithfully and 2–transitively.