Classifying groups with small squaring properties

Abstract

A group G is said to be a B(n, k) group if |A²| ≤ k for any n-subset A of G. The B(2, 3) groups were previously classified by Freiman and the B(3, 6) groups were classified by Parmenter. In addition, the B(3, 8) groups were partially classified by Berkovich, Freiman and Praeger, and their work was later completed by Longobardi and Maj. In this thesis, we will classify the B(2, k) and B(3, k) groups for all other values of k (except B(3, 7) where partial results are obtained). We will also provide some results for higher values of n, including the classification of the B(4, 10) groups by Parmenter and some classifications for a general value of n by Berkovich and also Herzog, Longobardi and Maj.