Let k be an imaginary quadratic number field with Ck,2, the 2-Sylow subgroup of its ideal class group Ck, of rank 4. We show that k has infinite 2-class field tower for particular families of fields k, according to the 4-rank of Ck, the Kronecker symbols of the primes dividing the discriminant ∆k of k, and the number of negative prime discriminants dividing ∆k. In particular we show that if the...

In this paper we describe an algorithm that outputs the order and the structure, including generators, of the 2-Sylow subgroup of an elliptic curve over a finite field. To do this, we do not assume any knowledge of the group order. The results that lead to the design of this algorithm are of inductive type. Then a right choice of points allows us to reach the end within a linear number of succe...

Eric Lander conjectured that if G is an abelian group of order v containing a difference set of order n and p is a prime dividing v and n, then the Sylow p-subgroup of G cannot be cyclic. This paper verifies a version of this conjecture for k < 6500. A special case of this version is the non-existence of Menon-Hadamard-McFarland difference sets in 2-groups. We also give an algorithm that easily...

Abstract Let G be a finite Frobenius group of degree n . We show, by elementary means, that is power some prime p provided the rank $${\mathrm{rk}}(G)\le 3+\sqrt{n+1}$$ rk ( G ) ≤ 3 + <...

1. The dihedral subgroup is non-abelian. If a group G has the property that the squares of its operators constitute a given group then the direct product of G and any abelian group of order 2m and of type (1,1, 1, • • • ) has the same property. Such direct products will always be excluded in what follows. It has been noted that a necessary and sufficient condition that there is at least one gro...

We extend a result of E. Hrushovski and A. Pillay as follows. Let G be a finite subgroup of GL(n,F) where F is a field of characteristic p such that p is sufficiently large compared to n. Assume thatG is generated by p-elements. ThenG is a product of 25 of its Sylow p-subgroups. If G is a simple group of Lie type in characteristic p, the analogous result holds without any restriction on the Lie...

Let G be a finite group, p a prime, and P a Sylow p-subgroup of G. Several recent refinements of the McKay conjecture suggest that there should exist a bijection between the irreducible characters of p′-degree of G and the irreducible characters of p′-degree of NG(P ), which preserves field of values of correspondent characters (over the p-adics). This strengthening of the McKay conjecture has ...

This paper is devoted to the study of the volcanoes of l-isogenies of elliptic curves over a finite field, focusing on their height as well as on the location of curves across its different levels. The core of the paper lies on the relationship between the l-Sylow subgroup of an elliptic curve and the level of the volcano where it is placed. The particular case l = 3 is studied in detail, givin...

If Λ is an indecomposable, non maximal, symmetric order, then the idealizer of the radical Γ := Id(J(Λ)) = J(Λ)# is the dual of the radical. If Γ is hereditary then Λ has a Brauer tree (under modest additional assumptions). Otherwise ∆ := Id(J(Γ)) = (J(Γ)2)#. If Λ = ZpG for a p-group G 6= 1, then Γ is hereditary iff G ∼= Cp and otherwise [∆ : Λ] = p2|G/(G′Gp)|. For Abelian groups G, the length ...

The algorithm we develop outputs the order and the structure, including generators, of the -Sylow subgroup of the group of rational points of an elliptic curve defined over a finite field. To do this, we do not assume any knowledge of the group order. We are able to choose points in such a way that a linear number of successive -divisions leads to generators of the subgroup under consideration....