we say that a finite group $g$ is conjugacy expansive if for anynormal subset $s$ and any conjugacy class $c$ of $g$ the normalset $sc$ consists of at least as many conjugacy classes of $g$ as$s$ does. halasi, mar'oti, sidki, bezerra have shown that a groupis conjugacy expansive if and only if it is a direct product ofconjugacy expansive simple or abelian groups.by considering a character analo...

in this work we consider conjugacy of elements and parabolic subgroups‎ ‎in details‎, ‎in a new class of artin groups‎, ‎introduced in an earlier work‎, ‎which may contain arbitrary parabolic subgroups‎. ‎in particular‎, ‎we find‎ ‎algorithmically minimal representatives of elements in a conjugacy class‎ ‎and also an algorithm to pass from one minimal representative to the others‎.

‎‎in this paper we determine all finite $2$-groups of‎ ‎class $2$ in which every automorphism of order $2$ leaving the frattini subgroup elementwise fixed is inner‎.

We correct the result of a previous paper which purported to show that L2(41) was not a subgroup of M. Our main result is that there is exactly one conjugacy class of subgroups L2(41) in the Monster. Such subgroups are self-normalizing and maximal. This leads to a new unexplained Moonshine phenomenon.

We prove that a knowledge of the character degrees of a finite group G and of their multiplicities determines whether G has a Sylow p-subgroup as a direct factor. An analogous result based on a knowledge of the conjugacy class sizes was known. We prove variations of both results and discuss their similarities.

A family of polynomials parameterized by the conjugacy classes of a finite Coxeter group is investigated. These polynomials, together with the character table of the group, determine the associated generic degrees. The polynomials are described completely for classes that meet a parabolic subgroup whose components are of type A or are dihedral, and for the class of Coxeter elements.

We show that, given a saturated fusion system, it is, under certain conditions, possible to identify SL2(q) acting on a natural module inside the normalizer of an essential subgroup. In particular, this is the case if the fusion system is non-constrained and has only one conjugacy class of essential subgroups.

In this article, a Blackburn group refers to a finite non-Dedekind group for which the intersection of all nonnormal subgroups is not the trivial subgroup. By completing the arguments of M. Hertweck, we show that all conjugacy class preserving automorphisms of Blackburn groups are inner automorphisms. 2000 Mathematics subject classification: primary 20D45; secondary 16S34.

Abstract A group G is said to be an FC -group if every conjugacy class of has finite order and a T subnormal subgroup normal in . In this paper we give characterization groups that are both -groups go further the study with few non-normal subgroups sense chain conditions. The structure satisfying maximal, minimal double condition on will described.

We construct a CAT(0) group containing a finitely presented subgroup with infinitely many conjugacy classes of finiteorder elements. Unlike previous examples (which were based on right-angled Artin groups) our ambient CAT(0) group does not contain any rank 3 free abelian subgroups. We also construct examples of groups of type Fn inside mapping class groups, Aut(Fk), and Out(Fk) which have infin...