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A note on simple medial quasigroups

K. K. Ščukin (1993)

Commentationes Mathematicae Universitatis Carolinae

A solvable primitive group with finitely generated abelian stabilizers is finite.

Extremely primitive groups and linear spaces

Haiyan Guan, Shenglin Zhou (2016)

Czechoslovak Mathematical Journal

A non-regular primitive permutation group is called extremely primitive if a point stabilizer acts primitively on each of its nontrivial orbits. Let 𝒮 be a nontrivial finite regular linear space and G Aut ( 𝒮 ) . Suppose that G is extremely primitive on points and let rank ( G ) be the rank of G on points. We prove that rank ( G ) 4 with few exceptions. Moreover, we show that Soc ( G ) is neither a sporadic group nor an alternating group, and G = PSL ( 2 , q ) with q + 1 a Fermat prime if Soc ( G ) is a finite classical simple group.

Finiteness results for Hilbert's irreducibility theorem

Peter Müller (2002)

Annales de l’institut Fourier

Let k be a number field, 𝒪 k its ring of integers, and f ( t , X ) k ( t ) [ X ] be an irreducible polynomial. Hilbert’s irreducibility theorem gives infinitely many integral specializations t t ¯ 𝒪 k such that f ( t ¯ , X ) is still irreducible. In this paper we study the set Red f ( 𝒪 k ) of those t ¯ 𝒪 k with f ( t ¯ , X ) reducible. We show that Red f ( 𝒪 k ) is a finite set under rather weak assumptions. In particular, previous results obtained by diophantine approximation techniques, appear as special cases of some of our results. Our method is different. We use elementary group...

Permutations which make transitive groups primitive

Pedro Lopes (2009)

Open Mathematics

In this article we look into characterizing primitive groups in the following way. Given a primitive group we single out a subset of its generators such that these generators alone (the so-called primitive generators) imply the group is primitive. The remaining generators ensure transitivity or comply with specific features of the group. We show that, other than the symmetric and alternating groups, there are infinitely many primitive groups with one primitive generator each. These primitive groups...

Polynomial Automorphisms Over Finite Fields

Maubach, Stefan (2001)

Serdica Mathematical Journal

It is shown that the invertible polynomial maps over a finite field Fq , if looked at as bijections Fn,q −→ Fn,q , give all possible bijections in the case q = 2, or q = p^r where p > 2. In the case q = 2^r where r > 1 it is shown that the tame subgroup of the invertible polynomial maps gives only the even bijections, i.e. only half the bijections. As a consequence it is shown that a set S ⊂ Fn,q can be a zero set of a coordinate if and only if #S = q^(n−1).

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