Abstract We prove that there exists a universal constant D such if p is prime divisor of the index Fitting subgroup finite group G , then number conjugacy classes at least $Dp/\log_2p$ . conjecture we can take $D=1$ and for solvable groups, $D=1/3$

We prove that the divisor class group of any open Richardson variety in Grassmannian is trivial. Our proof uses Nagata’s criterion, localizing coordinate ring at a suitable set Plücker coordinates. these coordinates are prime elements by showing subscheme they define an positroid variety. results hold over field and integers.

We calculate admissible values of r such that a square-free polynomial with integer coefficients, no fixed prime divisor and irreducible factors of degree at most 3 takes infinitely many values that are a product of at most r distinct primes.

We deal with classes of prime ideals whose associated graded ring is isomorphic to the Rees algebra of the conormal module in order to describe the divisor class group of the Rees algebra and to examine the normality of the conormal module.

Translation planes of order q and spread in PG(3, q), where q is an odd prime power and q− 1 has a p-primitive divisor, that admit a linear Abelian group of order (q + 1) containing at most three kernel homologies are shown to be associated to flocks of quadratic cones.

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. Then the global tame degree t(H) equals zero if and only if H is factorial (equivalently, |G| = 1). If |G| > 1, then D(G) ≤ t(H) ≤ 1 + D(G) (

In this thesis we look at several problems that lie in the intersection between combinatorial and multiplicative number theory. A common theme of many of these problems are estimates for and properties of the smooth numbers, those integers not divisible by any large prime factors. The first is the Ramsey-theoretic problem to determine the maximal size of a subset of the integers containing no 3...

Let a be an integer and q a prime number. In this paper we find an asymptotic formula for the number of positive integers n ≤ x with the property that no divisor d > 1 of n lies in the arithmetic progression a modulo q. MSC numbers: 11A05, 11A25, 11N37

Let M be a commutative cancellative monoid. The set ∆(M), which consists of all positive integers which are distances between consecutive irreducible factorization lengths of elements in M , is a widely studied object in the theory of nonunique factorizations. If M is a Krull monoid with divisor class group Zn, then it is well-known that ∆(M) ⊆ {1, 2, . . . , n − 2}. Moreover, equality holds fo...

We describe a new deterministic algorithm for the problem of constructing k th power nonresidues in nite elds GF p where p is prime and k is a prime divisor of p We prove under the assumption of the Extended Riemann Hypothesis ERH that for xed n and p our algorithm runs in polynomial time Unlike previous algorithms for this problem this polynomial time bound holds even if k is very large More g...