The Experts below are selected from a list of 24459 Experts worldwide ranked by ideXlab platform
P. C. Wong - One of the best experts on this subject based on the ideXlab platform.
-
Cyclic Conjugacy separability and Conjugacy separability of certain HNN extensions
Communications in Algebra, 2020Co-Authors: Kok Bin Wong, P. C. WongAbstract:In this note, we give a criterion for certain HNN extensions of cyclic Conjugacy separable (respectively Conjugacy separable) groups with infinite cyclic associated subgroups to be again cyclic con...
-
Cyclic Conjugacy separability of HNN extensions of cyclic Conjugacy separable groups
2016Co-Authors: K. B. Wong, P. C. WongAbstract:A group G is said to be cyclic Conjugacy separable (c.c.s.) if for each x ∈ G and each cyclic subgroup 〈y〉 of G such that no conjugate of x in G lies in 〈y〉, then there exists a finite homomorphic image G¯ of G such that no conjugate of x¯ in G¯ lies in 〈y¯〉. In this paper, we show that certain HNN extensions of cyclic Conjugacy separable groups are cyclic Conjugacy separable. We then apply our results to HNN extensions of polycyclic-by-finite groups.
-
Cyclic Conjugacy separability and Conjugacy separability of certain amalgamated free products
2015Co-Authors: K. B. Wong, P. C. WongAbstract:In this note, we study cyclic Conjugacy separability and Conjugacy separability of generalized free products. We first prove the criterions for generalized free products to be cyclic Conjugacy separable and Conjugacy separable. Our method involved a concept first introduced by G. Baumslag, called filterations. This method makes our criterions clearer and easier for applications. We then extend our criterions to generalized free products of polycyclic-by-finite groups or surface groups amalgamating a central subgroup. Finally, we show that generalized free products of free groups or finitely generated torsion-free nilpotent groups amalgamating an infinite cyclic subgroup are both cyclic Conjugacy separable and Conjugacy separable.
-
Criteria for Conjugacy separability and residual finiteness
2015Co-Authors: K. B. Wong, P. C. WongAbstract:The Conjugacy problem, word problem and related properties of groups like Conjugacy separability, cyclic Conjugacy separability and weak potency are areas of active research in Group Theory. In recent years, researchers extended this research to public-key cryptography. In this note, we proved criterions for extending the properties of cyclic Conjugacy separability, Conjugacy separability and weak potency to free products with amalgamation of polycyclic-by-finite and free-by-finite groups.
-
Conjugacy Separability of Certain HNN Extensions of Conjugacy-Separable Groups
Algebra Colloquium, 2000Co-Authors: P. C. Wong, C. K. TangAbstract:A group G is said to be Conjugacy-separable if, for each pair of elements x, y ɛ G such that x and y are not conjugate in G, there exists a finite homomorphic image G¯ of G such that the images of x and y are not conjugate in G¯. In this paper, we show that certain HNN extensions of Conjugacy-separable groups are Conjugacy-separable. We then apply our results to HNN extensions of polycyclic-by-finite groups.
George Lusztig - One of the best experts on this subject based on the ideXlab platform.
-
distinguished Conjugacy classes and elliptic weyl group elements
arXiv: Representation Theory, 2013Co-Authors: George LusztigAbstract:We define and study a correspondence between the set of distinguished G^0-Conjugacy classes in a fixed connected component of a reductive group G (with G^0 almost simple) and the set of (twisted) elliptic Conjugacy classes in the Weyl group. We also prove a homogeneity property related to this correspondence.
-
from Conjugacy classes in the weyl group to unipotent classes ii
arXiv: Representation Theory, 2011Co-Authors: George LusztigAbstract:Let G be an affine algebraic group over an algebraically closed field such that the identity component G^0 of G is reductive. Let W be the Weyl group of G and let D be a connected component of G whose image in G/G^0 is a unipotent element. In this paper we define a map from the set of "twisted conjugay classes" in W to the set of unipotent G^0-Conjugacy classes in D, generalizing an earlier construction which applied when G is connected.
Helena Verrill - One of the best experts on this subject based on the ideXlab platform.
-
symmetric groups and Conjugacy classes
Journal of Group Theory, 2008Co-Authors: Edith Adanbante, Helena VerrillAbstract:Let S n be the symmetric group of degree n where n > 5. Given any non-trivial , we prove that the product of the Conjugacy classes and is never a Conjugacy class. Furthermore, if n is odd and not a multiple of three, then is the union of at least three distinct Conjugacy classes. We also describe the elements in the case when is the union of exactly two distinct Conjugacy classes.
-
symmetric groups and Conjugacy classes
arXiv: Group Theory, 2007Co-Authors: Edith Adanbante, Helena VerrillAbstract:Let S_n be the symmetric group on n-letters. Fix n>5. Given any nontrivial $\alpha,\beta\in S_n$, we prove that the product $\alpha^{S_n}\beta^{S_n}$ of the Conjugacy classes $\alpha^{S_n}$ and $\beta^{S_n}$ is never a Conjugacy class. Furthermore, if n is not even and $n$ is not a multiple of three, then $\alpha^{S_n}\beta^{S_n}$ is the union of at least three distinct Conjugacy classes. We also describe the elements $\alpha,\beta\in S_n$ in the case when $\alpha^{S_n}\beta^{S_n}$ is the union of exactly two distinct Conjugacy classes.
Laura Ciobanu - One of the best experts on this subject based on the ideXlab platform.
-
The Conjugacy ratio of groups
arXiv: Group Theory, 2019Co-Authors: Laura Ciobanu, Armando MartinoAbstract:In this paper we introduce and study the Conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the Conjugacy and standard growth functions. We conjecture that the Conjugacy ratio is $0$ for all groups except the virtually abelian ones, and confirm this conjecture for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups, and the lamplighter group.
-
formal Conjugacy growth in acylindrically hyperbolic groups
International Mathematics Research Notices, 2016Co-Authors: Yago Antolin, Laura CiobanuAbstract:Rivin conjectured that the Conjugacy growth series of a hyperbolic group is rational if and only if the group is virtually cyclic. Ciobanu, Hermiller, Holt and Rees proved that the Conjugacy growth series of a virtually cyclic group is rational. Here we present the proof confirming the other direction of the conjecture, by showing that the Conjugacy growth series of a non-elementary hyperbolic group is transcendental. We also present and prove some variations of Rivin's conjecture for commensurability classes and primitive Conjugacy classes. We then explore Rivin's conjecture for finitely generated acylindrically hyperbolic groups and prove a formal language version of it, namely that no set of minimal length Conjugacy representatives can be unambiguous context-free.
-
Conjugacy languages in groups
arXiv: Group Theory, 2014Co-Authors: Laura Ciobanu, Susan Hermiller, Derek F. Holt, Sarah ReesAbstract:We study the regularity of several languages derived from Conjugacy classes in a finitely generated group G for a variety of examples including word hyperbolic, virtually abelian, Artin, and Garside groups. We also determine the rationality of the growth series of the shortlex Conjugacy language in virtually cyclic groups, proving one direction of a conjecture of Rivin.
-
Conjugacy growth series and languages in groups
Transactions of the American Mathematical Society, 2013Co-Authors: Laura Ciobanu, Susan HermillerAbstract:In this paper we introduce the geodesic Conjugacy language and geodesic Conjugacy growth series for a finitely generated group. We study the effects of various group constructions on rationality of both the geodesic Conjugacy growth series and spherical Conjugacy growth series, as well as on regularity of the geodesic Conjugacy language and spherical Conjugacy language. In particular, we show that regularity of the geodesic Conjugacy language is preserved by the graph product construction, and rationality of the geodesic Conjugacy growth series is preserved by both direct and free products. 2010 Mathematics Subject Classification: 20F65, 20E45.
-
Conjugacy growth series and languages in groups
arXiv: Group Theory, 2012Co-Authors: Laura Ciobanu, Susan HermillerAbstract:In this paper we introduce the geodesic Conjugacy language and geodesic Conjugacy growth series for a finitely generated group. We study the effects of various group constructions on rationality of both the geodesic Conjugacy growth series and spherical Conjugacy growth series, as well as on regularity of the geodesic Conjugacy language and spherical Conjugacy language. In particular, we show that regularity of the geodesic Conjugacy language is preserved by the graph product construction, and rationality of the geodesic Conjugacy growth series is preserved by both direct and free products.
Edith Adan-bante - One of the best experts on this subject based on the ideXlab platform.
-
On Conjugacy Classes of \\SL(2,q)
2012Co-Authors: Edith Adan-bante, John M. HarrisAbstract:Let \SL(2,q) be the group of 2\times 2 matrices with determinant one over a finite field F of size q. We prove that if q is even, then the product of any two noncentral Conjugacy classes of \SL(2,q) is the union of at least q-1 distinct Conjugacy classes of \SL(2,q). On the other hand, if q>3 is odd, then the product of any two noncentral Conjugacy classes of \SL(2,q) is the union of at least \fracq+32 distinct Conjugacy classes of \SL(2,q).
-
On Conjugacy classes of SL$(2,q)$
arXiv: Group Theory, 2009Co-Authors: Edith Adan-bante, John M. HarrisAbstract:Let SL(2,q) be the group of 2X2 matrices with determinant one over a finite field F of size q. We prove that if q is even, then the product of any two noncentral Conjugacy classes of SL(2,q) is the union of at least q-1 distinct Conjugacy classes of SL(2,q). On the other hand, if q>3 is odd, then the product of any two noncentral Conjugacy classes of SL(2,q) is the union of at least (q+3)/2 distinct Conjugacy classes of SL(2,q).
-
HOMOGENEOUS PRODUCTS OF Conjugacy CLASSES
Archiv der Mathematik, 2006Co-Authors: Edith Adan-banteAbstract:Let G be a finite group and a ∈G. Let a G ={g −1 a g | g ∈ G} be the Conjugacy class of a in G. Assume that a G and b G are Conjugacy classes of G with the property that C G (a)=C G (b). Then a G b G is a Conjugacy class if and only if [a,G]=[b,G]=[ab,G] and [ab,G] is a normal subgroup of G.
-
On nilpotent groups and Conjugacy classes
arXiv: Group Theory, 2005Co-Authors: Edith Adan-banteAbstract:Let $G$ be a nilpotent group and $a\in G$. Let $a^G=\{g^{-1}ag\mid g\in G\}$ be the Conjugacy class of $a$ in $G$. Assume that $a^G$ and $b^G$ are Conjugacy classes of $G$ with the property that $|a^G|=|b^G|=p$, where $p$ is an odd prime number. Set $a^G b^G=\{xy\mid x\in a^G, y\in b^G\}$. Then either $a^G b^G=(ab)^G$ or $a^G b^G$ is the union of at least $\frac{p+1}{2}$ distinct Conjugacy classes. As an application of the previous result, given any nilpotent group $G$ and any Conjugacy class $a^G$ of size $p$, we describe the square $a^G a^G$ of $a^G$ in terms of Conjugacy classes of $G$.
