Let q be a power of a prime and n a positive integer. Let P (q) be a parabolic subgroup of the finite general linear group GLn(q). We show that the number of P (q)conjugacy classes in GLn(q) is, as a function of q, a polynomial in q with integer coefficients. This answers a question of J. Alperin in [1].

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 + <...

SOME systems of collineation groups which arise in connection with the theory of elliptic functions have been investigated by Klein | and HurwitzJ. One of them is a system in n variables each group of which contains an invariant subgroup of order n. For n SL prime the quotient group with respect to this invariant subgroup is (1, 1) isomorphic with the modular group on two indices of order n(n —...

A new digital signature scheme and public key cryptosystem are proposed which use operations in a prime order subgroup of Z n for a composite number n. There are similarities with the best known digital signatures and public key cryptosystems (RSA and discrete logarithm based schemes) in terms of the mathematical structure. With regard to computational requirements the new schemes are competiti...

Let p be an odd prime number and F a field containing a primitive pth root of unity. We prove a new restriction on the group-theoretic structure of the absolute Galois group GF of F . Namely, the third subgroup G (3) F in the descending p-central sequence of GF is the intersection of all open normal subgroups N such that GF /N is 1, Z/p , or the extra-special group Mp3 of order p and exponent p.

Let G be a finite group and suppose that the set of conjugacy class sizes of G is f1;m;mng, where m; n > 1 are coprime. We prove that m 1⁄4 p for some prime p dividing n 1. We also show that G has an abelian normal p-complement and that if P is a Sylow p-subgroup of G, then jP 0j 1⁄4 p and jP=ZðGÞpj 1⁄4 p. We obtain other properties and determine completely the structure of G.

The purpose of this paper is to justify the claim that a method for generating primes presented at EUROCRYPT'89 generates primes with virtually uniform distribution. Using convincing heuristic arguments, the conditional probability distributions of the size of the largest prime factor p 1 (n) of a number n on the order of N is derived, given that n satisses one of the conditions 2n+1 is prime, ...

Let p be a prime and K a number field of degree p. We count the number of elliptic curves, up to K-isomorphism, having a prescribed property, where this property is either that the curve contains a fixed torsion group as a subgroup, or that it has an isogeny of prescribed degree. We also study the following question: for a given n such that |Y0(n)(Q)| > 0, does every elliptic curve over K with ...