Jack Sonn

Sylow-metacyclic groups andQ-admissibility

A finite groupG isQ-admissible if there exists a division algebra finite dimensional and central overQ which is a crossed product forG. AQ-admissible group is necessarily Sylow-metacyclic (all its Sylow subgroups are metacyclic). By means of an investigation into the structure of Sylow-metacyclic groups, the inverse problem (is every Sylow-metacyclic groupQ-admissible?) is essentially reduced to groups of order 2a 3b and to a list of known “almost simple” groups.

Rational division algebras as solvable crossed products

LetG be a finite group. If there exists a division algebra central over the rationalsQ which is a crossed product forG, then according to a theorem of Schacher, the Sylow subgroups ofG are all metacyclic. The converse is proved here to hold in the following cases: (1)G metacyclic; (2) The Sylow 2-subgroups ofG are cyclic (this impliesG solvable); (3)G is solvable and the Sylow 2-subgroups ofG are dihedral of order larger than 8.

BRAUER GROUPS, EMBEDDING PROBLEMS, AND NILPOTENT GROUPS AS GALOIS GROUPS*

Let ℚab denote the maximal abelian extension of the rationals ℚ, and let ℚabnil denote the maximal nilpotent extension of ℚab. We prove that for every primep, the free pro-p group on countably many generators is realizable as the Galois group of a regular extension of ℚabnil(t). We also prove that ℚabnil is not PAC (pseudo-algebraically closed).

On Galois Realizations of the 2-Coverable Symmetric and Alternating Groups

Let f(x) be a monic polynomial in ℤ[x] with no rational roots but with roots in ℚ p for all p, or equivalently, with roots mod n for all n. It is known that f(x) cannot be irreducible but can be a product of two or more irreducible polynomials, and that if f(x) is a product of m > 1 irreducible polynomials, then its Galois group must be “m-coverable”, i.e., a union of conjugates of m proper subgroups, whose total intersection is trivial. We are thus led to a variant of the inverse Galois problem: given an m-coverable finite group G, find a Galois realization of G over the rationals ℚ by a polynomial f(x) ∈ ℤ[x] which is a product of m nonlinear irreducible factors (in ℚ[x]) such that f(x) has a root in ℚ p for all p. The minimal value m = 2 is of special interest. It is known that the symmetric group S n is 2-coverable if and only if 3 ≤ n ≤ 6, and the alternating group A n is 2-coverable if and only if 4 ≤ n ≤ 8. In this article, we solve the above variant of the inverse Galois problem for the 2-coverable symmetric and alternating groups, and exhibit an explicit polynomial for each group, with the help of the software package GAP.

On the minimal ramification problem for ℓ -groups

Let p be a prime number. It is not known if every finite p-group of rank n>1 can be realized as a Galois group over Q with no more than n ramified primes. We prove that this can be done for the family of finite p-groups which contains all the cyclic groups of p-power order, and is closed under direct products, wreath products, and rank preserving homomorphic images. This family contains the Sylow p-subgroups of the symmetric groups and of the classical groups over finite fields of characteristic not p. On the other hand, it does not contain all finite p-groups.

Central extensions of Sn as Galois groups of regular extensions of Q(T)

Recently Mestre [Me] has proved that the double cover A,, of the alternating group A,, is realizable as the Galois group of a regular extension of the rational function field UJ( T), for all II 3 4. We show here that his method yields the same result for the two double covers S,: of the symmetric group S,Z, and more generally, for every finite central extension of S,. Partial results on S,, have been obtained previously by several authors [KSS, Sol, So2, SS, V].

Epimorphisms of demushkin groups

A necessary condition that a continuous epimorphism from a Demushkin groupG onto a finitep-groupH can be factored epimorphically through a free prop-groupS-is given, which is sufficient whenH is abelian of exponentpm≠2,m depending onG, 1≦m≦∞. In particular a free prop-factor groupS toG can have rank at most one half rankG. Application is made to embedding problems over localp-adic fields.

Relative Brauer groups and m-torsion

Let K be a field and Br(K) its Brauer group. If L/K is a field extension, then the relative Brauer group Br(L/K) is the kernel of the restriction map res L/K : Br(K) → Br(L). A subgroup of Br(K) is called an algebraic relative Brauer group if it is of the form Br(L/K) for some algebraic extension L/K. In this paper, we consider the m-torsion subgroup Br m (K) consisting of the elements of Br(K) killed by m, where m is a positive integer, and ask whether it is an algebraic relative Brauer group. The case K = Q is s already interesting: the answer is yes for m squarefree, and we do not know the answer for m arbitrary. A counterexample is given with a two-dimensional local field K = k((t)) and m = 2.

Class groups and Brauer groups

LetF be a global field,n a positive integer not divisible by the characteristic ofF. Then there exists a finite extensionE ofF whose class group has a cyclic direct summand of ordern. This theorem, in a slightly stronger form, is applied to determine completely, on the basis of the work of Fein and Schacher, the structure of the Brauer group Br(F()) of the rational function fieldF(t). As a consequence of this, an additional theorem of the above authors, together with a note at the end of the paper, imply that Br(F(t)) ≊ Br(F(t1, ···,tn)), wheret1, ···,tn are algebraically independent overF.

Central extensions of symmetric groups as Galois groups

’ Supported in part by the Wolf Foundation. *Supported in part by NSF Grant DMS85-00929. During this work, Murray Schacher was also a Fulbright-Hayes Fellow in Belgium under the auspices of the Commission for Educational Exchange Between the United States, Belgium, and Luxembourg.