### Abstract

The large Conway simple group Co1${{\rm Co}_{1}}$ contains a copy of the alternating group A9${{\rm A}_{9}}$ and thus contains a nested sequence A3≤A4≤…≤A9${{\rm A}_{3}\leq{\rm A}_{4}\leq\dots\leq{\rm A}_{9}}$. Shortly after Co1${{\rm Co}_{1}}$ was discovered, J. G. Thompson recognised that the normalizer of each of the groups in this sequence (apart from that of A8${{\rm A}_{8}}$) is maximal in Co1${{\rm Co}_{1}}$ and the resulting collection of subgroups 3Suz: 2$3\mathord{{}^{\;\textstyle{\cdot}}}{\rm Suz\,{:}\,2}$, (A4×G2(4)): 2$({\rm A}_{4}\times{\rm G}_{2}(4))\,{:}\,2$, (A5×HJ): 2$({\rm A}_{5}\times{\rm HJ})\,{:}\,2$, (A6×U3(3)): 2$({\rm A}_{6}\times{\rm U}_{3}({3}))\,{:}\,2$, (A7×L2(7)): 2$({\rm A}_{7}\times{\rm L}_{2}({7}))\,{:}\,2$, A8×S4${\rm A}_{8}\times{\rm S}_{4}$, A9×S3${\rm A}_{9}\times{\rm S}_{3}$ is now known as the Thompson chain , where Suz denotes the Suzuki simple group and HJ denotes the Hall–Janko group. Remarkably, we can start at the other end in the sense that if we consider U3(3)${{\rm U}_{3}({3})}$ in a certain way, we obtain a construction which produces each of the groups U3(3): 2${\rm U}_{3}({3})\,{:}\,2$, HJ: 2${\rm HJ}\,{:}\,2$, G2(4): 2${\rm G}_{2}(4)\,{:}\,2$, 3Suz: 2$3\mathord{{}^{\;\textstyle{\cdot}}}{\rm Suz}\,{:}\,2$, 2×Co1$2\times{\rm Co}_{1}$ spontaneously. Indeed, a presentation containing a parameter n is given which, for n=4,5,6,7${n=4,5,6,7}$, defines each of the above groups; n appears just twice in the presentation. Specifically, we associate with each directed edge ij of Kn${{\rm K}_{n}}$ (the complete graph on n vertices) an element tij${t_{ij}}$ of order 7 in some group G , where tji=tij-1${t_{ji}=t_{ij}^{-1}}$. We insist that G possesses automorphisms corresponding to the symmetric group permuting the n vertices of our Kn${{\rm K}_{n}}$, and in addition an automorphism which squares each of the tij${t_{ij}}$. If we now factor by a relation which ensures that a triangle generates U3(3)${{\rm U}_{3}({3})}$, then a K4${{\rm K}_{4}}$ generates HJ, a K5${{\rm K}_{5}}$ generates G2(4)${{\rm G}_{2}(4)}$, a K6${{\rm K}_{6}}$ generates 3Suz${3\mathord{{}^{\;\textstyle{\cdot}}}{\rm Suz}}$ and a K7${{\rm K}_{7}}$ generates Co1${{\rm Co}_{1}}$. What happens for n≥8${n\geq 8}$ is explained fully in the text. Thus this is not simply a sequence of nested subgroups in a larger group, but a finite family of closely-related perfect groups.