Given a finite group G by its multiplication table as input, we give a deterministic polynomial-time construction of a directed Cayley graph on G with O(log |G|) generators, which has a rapid mixing property and a constant spectral expansion. We prove a similar result in the undirected case, and give a new deterministic polynomialtime construction of an expanding Cayley graph with O(log |G|) ge...

We examine the minimal distance (number of differing entries) between different group tables of the same order n. Here group table means a matrix of order n with entries from a fixed set of n symbols, which (with suitable border elements) is the multiplication table of a group. (The border elements are not considered part of the table. A group is defined up to isomorphism by its multiplication ...

We consider the problem of determining if two finite groups are isomorphic. The groups are assumed to be represented by their multiplication tables. We present an O(n) algorithm that determines if two Abelian groups with n elements each are isomorphic. This improves upon the previous upper bound of O(n log n) [23] known for this problem. We solve a more general problem of computing the orders o...

A Latin square design whose automorphism group is transitive of rank at most 3 on points must come from the multiplication table of an elementary abelian p-group,

The Erdős multiplication table problem asks how many distinct integers appear in the N × table. order of magnitude this quantity was determined by Ford [8] . In paper we study number y -smooth entries table, that is to say with no prime factors greater than