Search

For every field $F$ which has a quadratic extension $E$ we show there are non-metabelian infinite-dimensional thin graded Lie algebras all of whose homogeneous components, except the second one, have dimension $2$. We construct such Lie algebras as $F$-subalgebras of Lie algebras $M$ of maximal class over $E$. We characterise the thin Lie $F$-subalgebras of $M$ generated in degree $1$. Moreover we show that every thin Lie algebra $L$ whose ring of graded endomorphisms of degree zero of $L^3$ is a quadratic extension of $F$ can be obtained in this Lie algebra of maximal class over $E$ which are ideally $r$-constrained for a positive integer $r$., 10 pages

In his controversial speech at the opening ceremony of the 2013 Frankfurt Book Fair, Luiz Ruffato proclaimed that the greatest dilemma human beings have ever faced is dealing with the dichotomy between self and other. He went on to point out how this dichotomy lies at the heart of Brazilian identity, indeed his first example of the clash between cultures that has gone on to shape his country’s identity and national ideology was the original encounter between the indigenous peoples and the European colonizers. Brazil, Ruffato declared, was born under the aegis of genocide. The subject arose again a year later in Ruffato’s column for the Spanish newspaper El País, where he further highlighted some shameful facts about the disgraceful way that indigenous people are treated in Brazil, including the horrific data that according to a UNICEF study the rate of suicide among the indigenous (36% of whom are aged under 14) was four times higher than that of the rest of the population, and in some places as much as 34 times higher. Ruffato’s signalling of the plight of the indigenous peoples as well as recognizing their effective lack of visibility despite their fundamental place in Brazilian national identity echoes the protests made directly by indigenous groups or their representatives for their rights as citizens to be respected, especially their rights to ancestral lands, which are so often ignored. It also reminds readers of the symbolic power of the indigenous people, their portrayal (romanticized, demonized, hyperbolized) almost always skewed to fit a particular agenda and almost always clashing with reality. Essentialised and caricatured in literature, art, film, history, anthropology and politics, indigenous peoples paradoxically represent numerous contradictory elements in the Brazilian national ideology: they are savage yet passive, both backward and wise, the emblematic and incompatible Other who is also an ancestor.

In his controversial speech at the opening ceremony of the 2013 Frankfurt Book Fair, Luiz Ruffato proclaimed that the greatest dilemma human beings have ever faced is dealing with the dichotomy between self and other. He went on to point out how this dichotomy lies at the heart of Brazilian identity, indeed his first example of the clash between cultures that has gone on to shape his country’s identity and national ideology was the original encounter between the indigenous peoples and the European colonizers. Brazil, Ruffato declared, was born under the aegis of genocide. The subject arose again a year later in Ruffato’s column for the Spanish newspaper El País, where he further highlighted some shameful facts about the disgraceful way that indigenous people are treated in Brazil, including the horrific data that according to a UNICEF study the rate of suicide among the indigenous (36% of whom are aged under 14) was four times higher than that of the rest of the population, and in some places as much as 34 times higher. Ruffato’s signalling of the plight of the indigenous peoples as well as recognizing their effective lack of visibility despite their fundamental place in Brazilian national identity echoes the protests made directly by indigenous groups or their representatives for their rights as citizens to be respected, especially their rights to ancestral lands, which are so often ignored. It also reminds readers of the symbolic power of the indigenous people, their portrayal (romanticized, demonized, hyperbolized) almost always skewed to fit a particular agenda and almost always clashing with reality. Essentialised and caricatured in literature, art, film, history, anthropology and politics, indigenous peoples paradoxically represent numerous contradictory elements in the Brazilian national ideology: they are savage yet passive, both backward and wise, the emblematic and incompatible Other who is also an ancestor.

A complete map for a group G is a permutation phi: G --> G such that g bar right arrow g phi (g) is still a permutation of G. A conjecture of M. Hall and L. J. Paige states that every finite group whose Sylow 2-subgroup is non-trivial and non-cyclic admits a complete map. In the present paper it is proved that a potential counterexample G of minimal order to this conjecture either is almost simple or G has only one involution, the Sylow 2-subgroups of G are quaternionic, \G/G\ less than or equal to 2, G' congruent to SL(2, q) for some odd prime power q > 5 and if G is not a perfect group then G/Z(G') congruent to PGL(2, q).

A bijective mapping φ: G → G defined on a finite group G is complete if the mapping η defined by η(x) = xφ(x), x ∈ G, is bijective. In 1955 M. Hall and L. J. Paige conjectured that a finite group G has a complete mapping if and only if a Sylow 2-subgroup of G is non-cyclic or trivial. This conjecture is still open. In this paper we construct a complete mapping for the projective groups PSL(2, q), q ≡ 1 mod 4 and PGL(2, q), q odd. As a consequence, we prove that in odd characteristic the projective groups PGL(n, q), n > 2, and the linear groups GL(n, q), n ≥ 2, admit a complete mapping.