On A Combinatorial Problem In Group Theory

A combinatorial problem of group theory is considered in this paper.Let G be a group.Let A={a1,a2,...,an} be an n-element subset of G and we define A2={aiaj|1≤i,j≤n}.It is obvious that |A2|≤n(n+1)/2 if G is abelian.A group G is said to be a Bn-group if |A2|≤n(n+1)/2 for any n-element subset A of G[1].Therefore,Bn-groups can be viewed as a generalization of the abelian groups.In[2],the definition of small squaring property was proposed by Berkovich,Freiman and Praeger:a group G is called to have small squaring property if |A2|<n2 for any n-subset A of G.These notions were generalized to B(n,k)groups in[3]by Eddy and Parmenter:A finite group G is called a B(n,k)group if |A2|≤k for any n-subset A,where k is an integer with n≤k≤n2-1.It is obvious to see that a Bn-group is just a B(n,n(n+1)/2)group and a group with small squaring property is a B(n,n2-1)group.A B(n,k)group G is called trivial if |G|≤k.It is an interesting question to classify the nonabelian nontrivial B(n,k)groups for n(n+1)/2≤k≤n2-1.In this paper,the general relations between B(n+1,n+k)groups and B(n,k)groups are found by a combinatorial method.Using these relations,a class of B(n,k)groups are characterized by combining the known conclusions about B(n,k)groups and the computer software GAP.Specifically,the classifications of B(6,22),B(6,23),B(6,24),B(7,29),B(7,30)B(8,36),B(8,37)and B(9,45)groups are given in this paper,which enriches the theory of B(n,k)groups.