A new family of non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation is constructed. Two subfamilies, consisting of irretractable square-free solutions, are new counterexamples to Gateva-Ivanova's Strong Conjecture. They are in addition to those obtained by Vendramin.

Questi appunti nascono da corsi e cicli di seminari tenuti dagli autori nell'Università del Salento e nell'Università di Napoli Federico II e da numerose conversazioni tra gli autori stessi su una teoria - quella dei gruppi con classi di coniugio finite - che a distanza di settanta anni dalla sua nascita rimane vitale e ricca di problemi ancora da risolvere. La trattazione non ha pretese di completezza, ma vuole fornire gli strumenti essenziali della teoria ed alcuni suoi recenti sviluppi, suggerendo possibili ulteriori ricerche.

In this paper, we produce a method to construct quasi-linear left cycle sets A with Rad(A) contained in Fix(A).

In the present note, we show that if G is a finite group ad (FG)^+ is Lie metabelian, then G is nilpotent. Based on this result, we deduce that id G i torsion, pneq 3 and (FG)^+ is Lie metabelian, then G must abelian. This extends a result of Levin and Rosemberger.

Let A be a non-trivial semiprime associative superalgebra with superinvolution. In the present note we investigate when the subspaces of symmetric elements or skew elements of A are Lie nilpotent or Lie solvable. We show that these conditions determine the algebraic stucture of A

The central kernel K(G) of a group G is the subgroup consisting of all elements fixed by every central automorphism of G. It is proved here that if G is a finite-by- nilpotent group whose central kernel has finite index, the G is finite over the centre, and the elements of finite order of G form a finite subgroup; in particular G is finite, provided that it is periodic. Moreover, if G is a periodic finite-b-nilpotent group and G/K(G) is a Cernikov group, it turns out that G itself is a Cernikov group.

The first author and Rizzo established a link between regular subgroups of the affine group and radical brace over a field on the underlying vector space. We propose new constructions of radical braces that allow to obtain rather systematic construction of regular subgroups of the affine group. In particular, this approach allows to put in a more general context the regular subgroups constucted by C. Tamburini Bellani.

In this paper we prove some results about the derived length of the unit group of a group algebra. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

In this paper we introduce the asymmetric product of radical braces, a construction which extends the semidirect product of radical braces. This new construction allows to obtain rather systematic constructions of regular subgroups of the affine group and, in particular, our approach allows to put in more general context the regular subgroups constructed by Hegedus.

Let L be a non-abelian restricted Lie algebra over a field of characteristic p>0 and let u(L) denote its restricted enveloping algebra. In 2006 it was proved that if u(L) is Lie solvable then the Lie derived length of u(L) is at least $lceillog_2(p+1)rceil$. In the present paper we characterize the restricted enveloping algebras whose Lie derived length coincides with this lower bound.

In this paper we obtain new solutions of the Yang-Baxter equation that are left non-degenerate throught left semi-braces, a generalization of braces introduced by Rump. In order to provide new solution we introduce the asymmetric product of left semi-braces, a genralization of the semidirect product of braces, that allow us to produce several examples of left semi-braces.

An associative ring R, not necessarily with an identity element, is called semilocal if R modulo its Jacobson radical is an artinian ring. It is proved that if the adjoint group of a semilocal ring R is locally supersoluble, then R is locally Lie supersoluble and its Jacobson radical is contained in a locally Lie nilpotent ideal of finite index in R.

These notes come from courses and seminars given by the authors at the University of Salento and at the University of Napoli "Federico II", and from several conversations between the authors themselves on a theory - that of groups with finite conjugacy classes - that seventy years after its birth remains vital and full of problems still to be solved. There are two relevant monographs on the theory of FC-groups, one due to Y.M. Garcakov and the order to M.J. Tomkinson, but both of them go back to over thirty years ago, and since then many new contributions have brought new light to the existing problems and posed new relevant questions. Our treatment has no claim to completeness, but is intended to provide the essential tools to the theory and some of its recent developments, suggesting possible further research topics.

Let F be a field of characteristic different from 2 and G a group with involution*. Extend the involution to the group ring FG and write (FG)^- for the Lie subalgebra of FG consisting of the skew elements. We classify the torsion group G having no elements of order 2 such that (FG)^- is bounded Lie Engel.