‎there are a few finite groups that are determined up to isomorphism solely by their order, such as $mathbb{z}_{2}$ or $mathbb{z}_{15}$. still other finite groups are determined by their order together with other data, such as the number of elements of each order, the structure of the prime graph, the number of order components, the number of sylow $p$-subgroups for each prime $p$, etc. in this...

‎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 the ranks of all the conjugacy classes of‎ ‎elements for simple alternating group $A_{10}$ using the structure‎ ‎constants method and other results established in‎ ‎[A.B.M‎. ‎Bas...

‎let $g$ be a finite group‎. ‎an element $gin g$ is called non-vanishing‎, ‎if for‎ ‎every irreducible complex character $chi$ of $g$‎, ‎$chi(g)neq 0$‎. ‎the bi-cayley graph $bcay(g,t)$ of $g$ with respect to a subset $tsubseteq g$‎, ‎is an undirected graph with‎ ‎vertex set $gtimes{1,2}$ and edge set ${{(x,1),(tx,2)}mid xin g‎, ‎ tin t}$‎. ‎let $nv(g)$ be the set‎ ‎of all non-vanishing element...

We construct an example of conjugacy separable group possessing a not conjugacy separable subgroup of finite index. We give also a sufficient condition for a conjugacy separable group to preserve this property when passing to subgroups of finite index. We establish also conjugacy separability of finitely presented residually free groups using impressive results of Bridson and Wilton [BW-07].

of distributions. The terms on the right are parametrized by "cuspidal automorphic data", and are defined in terms of Eisenstein series. They have been evaluated rather explicitly in [3]. The terms on the left are parametrized by semisimple conjugacy classes and are defined in terms of related G(A) orbits. The object of this paper is to evaluate these terms. In previous papers we have already e...

In this note, we give characterizations for certain HNN extensions of polycyclic-by-finite groups with central associated subgroups to be subgroup separable and conjugacy separable. We shall do this by showing the equivalence of subgroup separability and conjugacy separability in this type of HNN extensions. 2000 Mathematics Subject Classification: Primary 20E06, 20E26; Secondary 20F10, 20F19

We describe the isomorphism class of the torus centralizing a regular, semi-simple, stable conjugacy class in a simply-connected, semi-simple group. Let k be a field, and let G be a semi-simple, simply-connected algebraic group, which is quasi-split over k. The theory of semi-simple conjugacy classes in G is well understood, from work of Steinberg [S] and Kottwitz [K]. Any semi-simple conjugacy...

— Let r > 1 be a real. In this paper, we study piecewise class P Cr circle homeomorphisms with irrational rotation numbers. We give characterizations for such homeomorphisms that are piecewise Cr conjugate to Cr diffeomorphisms. As a consequence, we obtain a criterion of piecewise Cr conjugacy to diophantine rotations. This characterization extends those obtained by Liousse for the PL circle ho...

‎let $g$ be a finite group and $z(g)$ be the center of $g$‎. ‎for a subset $a$ of $g$‎, ‎we define $k_g(a)$‎, ‎the number of conjugacy classes of $g$ which intersect $a$ non-trivially‎. ‎in this paper‎, ‎we verify the structure of all finite groups $g$ which satisfy the property $k_g(g-z(g))=5$ and classify them‎.

many results were proved on the structure of finite groups with some‎ ‎restrictions on their real elements and on their conjugacy classes‎. ‎we‎ ‎generalize a few of these to some classes of infinite groups‎. ‎we study groups in which real elements are central‎, ‎groups in which real elements are $2$-elements‎, ‎groups in which all non-trivial classes have the same finite size and $fc$-groups w...