A nilpotent group G is a finite group that is the direct product of its Sylow p-subgroups. Theorem 1.1 (Fitting's Theorem) Let G be a finite group, and let H and K be two nilpotent normal subgroups of G. Then HK is nilpotent. Hence in any finite group there is a unique maximal normal nilpotent subgroup, and every nilpotent normal subgroup lies inside this; it is called the Fitting subgroup, and...

A reactive system is a system that maintains interactions with its environment. Therefore it is necessary for realizability[1][2] of its specification that the reactive system can keep reacting correctly to any kind of its environment behavior. Temporal logics have been studied as a language for describing reactive system specifications[3][4] and a realizability judgment method for temporal spe...

We present a generalization to genus 2 of the probabilistic algorithm in Sutherland [28] for computing Hilbert class polynomials. The improvement over the algorithm presented in [5] for the genus 2 case, is that we do not need to find a curve in the isogeny class with endomorphism ring which is the maximal order: rather we present a probabilistic algorithm for “going up” to a maximal curve (a c...

We present a generalization to genus 2 of the probabilistic algorithm of Sutherland for computing Hilbert class polynomials. The improvement over the Bröker-Gruenewald-Lauter algorithm for the genus 2 case is that we do not need to find a curve in the isogeny class whose endomorphism ring is the maximal order; rather, we present a probabilistic algorithm for “going up” to a maximal curve (a cur...

Let F be the finite field of order q, q = p with p prime. It is commonly atribute to J.P. Serre the fact that any curve F-covered by the Hermitian curve Hq+1 : y = x + x is also F-maximal. Nevertheless, the converse is not true as the GiuliettiKorchmáros example shows provided that q > 8 and h ≡ 0 (mod 3). In this paper, we show that if an F-maximal curve X of genus g ≥ 2 where q = p is such th...

The genus g of an Fq2-maximal curve satisfies g = g1 := q(q − 1)/2 or g ≤ g2 := ⌊(q − 1) /4⌊. Previously, Fq2 -maximal curves with g = g1 or g = g2, q odd, have been characterized up to Fq2 -isomorphism. Here it is shown that an Fq2 -maximal curve with genus g2, q even, is Fq2 -isomorphic to the nonsingular model of the plane curve ∑t i=1 y q/2 = x, q = 2, provided that q/2 is a Weierstrass non...