We obtain the following characterization of the solvable radicalR(G) of any finite group G: R(G) coincides with the collection of all g ∈ G such that for any 3 elements a1, a2, a3 ∈ G the subgroup generated by the elements g, aiga i , i = 1, 2, 3, is solvable. In particular, this means that a finite group G is solvable if and only if in each conjugacy class of G every 4 elements generate a solv...

Let G be a finite group with an abelian normal subgroup N . When does N have a unique conjugacy class of complements in G? We consider this question with a focus on subgroups and properties of maximal subgroups. As corollaries we obtain Theorems 1.6 and 1.7 which are closely related to a result by Parker and Rowley on supplements of a nilpotent normal subgroup (Theorem 1 of [3]). Furthermore, w...

Let S be a semigroup. The elements a, b ∈ S are called primarily conjugate if a = xy and b = yx for certain x, y ∈ S. The relation of conjugacy is defined as the transitive closure of the relation of primary conjugacy. In the case when S is a monoid, denote by G the group of units of S. Then the relation of G-conjugacy is defined by a ∼G b ⇐⇒ a = gbg for certain g ∈ G. We establish the structur...

By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. We conjecture that this measure agrees with a second measure on conjugacy classes of the Weyl group induced by a construction o...

We define a Rohlin property for Z-actions on UHF algebras and show a non-commutative Rohlin type theorem. Among those actions with the Rohlin property, we classify product type actions up to outer conjugacy. We consider two classes of UHF algebras. For UHF algebras in one class including the CAR algebra, there is one and only one outer conjugacy class of product type actions and for UHF algebra...

We solve the subgroup conjugacy problem for parabolic subgroups and Garside subgroups of a Garside group, and we present deterministic algorithms. This solution may be improved by using minimal simple elements. For standard parabolic subgroups of Garside groups we provide e ective algorithms for computing minimal simple elements.

We study the structure constants of the class algebra RZ(Γn) of the wreath products Γn associated to an arbitrary finite group Γ with respect to a basis provided by the conjugacy classes. A suitable filtration on the RZ(Γn) gives rise to the rings GΓ(n) with non-negative integer structure constants independent of n, which are then encoded in a single (Farahat-Higman) ring GΓ. We establish vario...

We present a new family of gauge invariant non-local order parameters ∆α for (non-abelian) discrete gauge theories on a Euclidean lattice, which are in one-to-one correspondence with the excitation spectrum that follows from the representation theory of the quantum double D(H) of the finite group H . These combine magnetic flux-sector labeled by a conjugacy class with an electric representation...