Counting squares of \(n\)-subsets in finite groups (Q2760990)

The authors consider the following problem from combinatorial group theory: For \(G\) a finite group and \(A=(a_{ij})\) an \(n\times n\), \(n\geq 2\), matrix with entries in \(G\), does there exist an \(n\)-tuple \(x=(x_1,\dots,x_n)\) of mutually distinct elements of \(G\) such that \(A\) is the multiplication table \(a_{ij}=x_ix_j\) of \(x\)? How many such tuples are there? Three theorems, A, B, and C, and several other results are proven. A: For every fixed \(A\) the number of corresponding \(x\)s is a multiple of \(|G|\). The case \(n=3\) is then investigated and all the 51 \(A\)s for which there is a \(G\) having a triple with the multiplication table \(A\) are described; these matrices were found first by \textit{G. A. Freiman} [Aequationes Math. 22, 140-152 (1981; Zbl 0489.20020)]. In the results B and C \(n=3\). B: \(G\) has only one \(3\times 3\) multiplication table iff \(G\) is cyclic of order 3, 4 or 5 or is an elementary Abelian \(2\)-group. C: If each \(3\times 3\) multiplication table of \(G\) contains at least two equal entries then \(G\) is soluble. In particular, if \(|G|\) is odd then \(G\) is Abelian, and if \(|G |\) is even then \(G\) has a normal Abelian \(2\)-Hall component.