Structure of \(R(3,3)\)-groups (Q1802327)

Let \(A = \{a_ 1,\dots,a_ m\}\) be a subset of a group \(G\) and put \(A^{[m]} = \{a_{\pi(1)}\dots a_{\pi(m)}: \pi \in S_ m\}\). \(G\) is called an \(R(m,n)\)-group if \(| A^{[m]}| \leq n\) for all \(m\)- element subsets \(A\) of \(G\). Having investigated the first non-trivial case \(R(3,2)\) earlier the authors now study the next one \(R(3,3)\). They derive the following theorem: A group \(G\) is an \(R(3,3)\)-group if and only if \(| G'| \leq 3\). More specifically an \(R(3,3)\)-group with \(| G'| = 1\) is abelian, with \(| G'| = 2\) is a non-abelian \(R(3,2)\)-group, and with \(| G'| = 3\) fulfills either \(\exp(G/Z(G)) = 3\) or \(G/Z(G) \cong S_ 3\). Connections with totally \(n\)-rewriteable semigroups are glanced at.