Generalized conjugation is the action of a group on its underlying set given by (g, x) 7→ φ(g)xg, where φ : G → G is some fixed endomorphism. Here we study combinatorial properties of the sizes of the orbits of the preceding action. In particular, we reduce the problem to a simpler case if φ has a nontrivial kernel or if it is an inner automorphism, and we give a construction that allows a part...

fullerene chemistry is nowadays a well-established field of both theoretical and experimental investigations‎. this study considers the symmetry of small fullerenes cage c24 and c28‎. ‎using pm3 program for c24 and c28 fullerenes, oh and td symmetry were confirmed, respectively‎. ‎ the mentioned algorithm to compute the automorphism group of these fullerenes with connectivity and geometry of th...

0. Introduction 1105 1. Constructible families 1111 2. Genericity for finite subgroups 1116 3. Finite groups of Lie type 1119 4. Basic nonconcentration estimate 1122 5. Finite subgroups of abelian varieties 1126 6. Orders of conjugacy classes and centralizers 1127 7. Regular semisimple and unipotent elements 1129 8. Minimal unipotent elements 1134 9. Frobenius map 1140 10. Traces in the basic c...

We analyse the rack structure of conjugacy classes in simple Suzuki and Ree groups determine which are kthulhu. Combining these results with abelian techniques, we show that only finite-dimensional complex pointed Hopf algebras over their group algebras.

We define new subclasses of the class of irreducible sofic shifts. These classes form an infinite hierarchy where the lowest class is the class of almost finite type shifts introduced by B. Marcus. We give effective characterizations of these classes with the syntactic semigroups of the shifts. We prove that these classes define invariants shift equivalence (and thus for conjugacy). Finally, we...

It is studied how taking the inverse image by a sliding block code affects the syntactic semigroup of a sofic subshift. Two independent approaches are used: ζ-semigroups as recognition structures for sofic subshifts, and relatively free profinite semigroups. A new algebraic invariant is obtained for weak equivalence of sofic subshifts, by determining which classes of sofic subshifts naturally d...

Abstract We prove that the number of conjugacy classes a finite group G consisting elements odd order, is larger than or equal to for normaliser Sylow 2-subgroup . This predicted by Alperin Weight Conjecture.