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We compute all signatures of P S L 2 ( 7) and P S L 2 ( 1 1) which classify all orientation preserving actions of the groups P S L 2 ( 7) and P S L 2 ( 1 1) on compact, connected, orientable surfaces with orbifold genus ≥ 0. This classification is well-grounded in the other branches of Mathematics like topology, smooth and conformal geometry, algebraic categories, and it is also directly related to the inverse Galois problem. [ABSTRACT FROM AUTHOR]

The Schur covers of a dihedral group of 2-power order are a dihedral group, a semidihedral group, and a generalized quaternion group. We prove that if one of the three groups is realizable as Galois group, then other two groups are also realizable as Galois groups under a condition on the existence of a certain quartic cyclic extension over the base field. [ABSTRACT FROM AUTHOR]

A theorem of Chow concerns homomorphisms of two abelian varieties under a primary field extension base change. In this paper, we generalize Chow's theorem to semi-abelian varieties. This contributes to different proofs of a well-known result that every algebraic torus splits over a finite separable field extension. We also obtain the best bound for the degrees of splitting fields of tori. [ABSTRACT FROM AUTHOR]

Let U be an affine smooth curve defined over an algebraically closed field of positive characteristic. The Abhyankar conjecture (proved by Raynaud and Harbater in 1994) describes the set of finite quotients of Grothendieck's étale fundamental group π1ét(U). In this paper, we consider a purely inseparable analogue of this problem, formulated in terms of Nori's profinite fundamental group scheme πN(U), and give a partial answer to it. [ABSTRACT FROM AUTHOR]

We introduce splitting coverings to enhance the well known analogy between field extensions and covering spaces. Semi-topological Galois groups are defined for Weierstrass polynomials and a Galois correspondence is proved. Combining results from braid groups, we solve the topological inverse Galois problem. As an application, symmetric and cyclic groups are realized over ℚ. [ABSTRACT FROM AUTHOR]

We prove the existence of certain rationally rigid triples in ${E}_{8}(p)$ for good primes $p$ (i.e. $p>5$) thereby showing that these groups occur as Galois groups over the field of rational numbers. We show that these triples arise from rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic zero. As a byproduct of the proof, we derive a remarkable symmetry between the character table of a finite reductive group and that of its dual group. We also give a short list of possible overgroups of regular unipotent elements in simple exceptional groups. [ABSTRACT FROM PUBLISHER]

We study the inverse Galois problem with restricted ramifications. Let p and q be distinct odd primes such that p ≡ 1 mod q. Let E(p3) be the non-abelian group of order p3 such that the exponent is equal to p, and let k be a cyclic extension over Q of degree q. In this paper, we study the existence of unramified extensions over k with the Galois group E(p3). We also give some numerical examples computed with PARI. [ABSTRACT FROM AUTHOR]

By a celebrated theorem of Harbater and Pop, the regular inverse Galois problem is solvable over any field containing a large field. Using this and the Mordell conjecture for function fields, we construct the first example of a field K over which the regular inverse Galois problem can be shown to be solvable, but such that K does not contain a large field. The paper is complemented by model-theoretic observations on the diophantine nature of the regular inverse Galois problem. [ABSTRACT FROM AUTHOR]