In a previous investigation [1], the author has studied finite groups © of an order g = pgo where p is a prime and go an integer not divisible by p. This work has been continued by H. F. Tuan [51 Let t denote the number of conjugate classes of ® which consist of element of order p. Tuan dealt with the groups ® for which t^2 and which have a faithful representation of degree less than p 1. We sh...

Let SQ(d) be the set of primes p for which there exists a number field K of degree ≤ d and an elliptic curve E/Q, such that the order of the torsion subgroup of E(K) is divisible by p. In this article we give bounds for the primes in the set SQ(d). In particular, we show that, if p ≥ 11, p 6= 13, 37, and p ∈ SQ(d), then p ≤ 2d + 1. Moreover, we determine SQ(d) for all d ≤ 42, and give a conject...

Let K be a field and f(X) be a separable polynomial in K[X]. The Galois group of f(X) over K permutes the roots of f(X) in a splitting field, and labeling the roots as r1, . . . , rn provides an embedding of the Galois group into Sn. We recall without proof two theorems about this embedding. Theorem 1.1. Let f(X) ∈ K[X] be a separable polynomial of degree n. (a) If f(X) is irreducible in K[X] t...

Let G be a nite group, F a eld whose characteristic p divides the order of G and A G the invariant ring of a nite-dimensional FG-module V. In analogy to modular representation theory we deene for any subgroup H G the (relative) trace-ideal A G H /A G to be the image of the relative trace map t G X is always a proper ideal of A G ; in fact, we show that its height is bounded above by the codimen...

Combinatorial designs have been used to construct digital fingerprint codes. Here, a new constructive algorithm for an anticollusion fingerprint code based on group-divisible designs is presented. These codes are easy to construct and available for a large number of individuals, which is important from a business point of view. Group-divisible designs have not been used previously as a tool for...

A group is small if it has countably many pure n-types for each integer n. It is shown that in a small group, subgroups which are definable with parameters in a finitely generated algebraic closure satisfy local descending chain conditions. An infinite small group has an infinite abelian subgroup, which may not be definable. A nilpotent small group is the central product of a definable divisibl...