Let G be a complex semisimple Lie group and τ a complex antilinear involution that commutes with a Cartan involution. If H denotes the connected subgroup of τ-fixed points in G, and K is maximally compact, each H-orbit in G/K can be equipped with a Poisson structure as described by Evens and Lu. We consider sym-plectic leaves of certain such H-orbits with a natural Hamiltonian torus action. A s...

We recognize the Gromoll-Meyer sphere Σ as the geodesic join of a simple closed geodesic and a minimal subsphere Σ ⊂ Σ, which can be equivariantly identified with the Brieskorn sphere W 5 3 . As applications we in particular determine the full isometry group of Σ, classify all closed subgroups which act freely, determine the homotopy types of the corresponding orbit spaces, identify the Hirsch-...

In this note we give examples of division rings which posses an anti-automorphism but no involution. The motivation for such examples comes from geometry. If D is a division ring and V a finite-dimensional right D-vector space of dimension ≥ 3, then the projective geometry P(V ) has a duality (resp. polarity) if and only if D has an anti-automorphism (resp. involution) [2, p. 97, p. 111]. Thus,...

‎let $r$ be a $*$-prime ring with center‎ ‎$z(r)$‎, ‎$d$ a non-zero $(sigma,tau)$-derivation of $r$ with associated‎ ‎automorphisms $sigma$ and $tau$ of $r$‎, ‎such that $sigma$‎, ‎$tau$‎ ‎and $d$ commute with $'*'$‎. ‎suppose that $u$ is an ideal of $r$ such that $u^*=u$‎, ‎and $c_{sigma,tau}={cin‎ ‎r~|~csigma(x)=tau(x)c~mbox{for~all}~xin r}.$ in the present paper‎, ‎it is shown that...

We study the existence of unitriangular basic sets for symmetric group which behave nicely with respect to Mullineux involution. Such give a natural labelling modular irreducible representations. show that, any odd prime p, p-blocks weight 2 have stable we describe by studying combinatorics partitions in these blocks.

Let ZG be the integral group ring of the finite nonabelian group G over the ring of integers Z, and let ∗ be an involution of ZG that extends one of G. If x and y are elements of G, we investigate when pairs of the form (uk,m(x), uk,m(x ∗)), or (uk,m(x), uk,m(y)), formed respectively by Bass cyclic and ∗-symmetric Bass cyclic units, generate a free noncyclic subgroup of the unit group of ZG. 1....

The result involution graph of a finite group , denoted by is an undirected simple whose vertex set the whole and two distinct vertices are adjacent if their product element. In this paper, graphs for all Mathieu groups connectivity in studied. diameter, radius girth also Furthermore, several other properties obtained.