## Old research snapshots

This file is a repository of research snapshots that have previously appeared on my web page.

#### Diagonal groups

Diagonal groups are one class of primitive permutation groups arising in the O'Nan–Scott theorem, and are slightly mysterious. With Rosemary Bailey, Cheryl Praeger and Csaba Schneider, I am working on understanding these groups better. As spin-offs, I have shown (with John Bray, Qi Cai, Pablo Spiga and Hua Zhang), using the Hall–Paige conjecture, that diagonal groups with at least three simple factors in the socle are non-synchronizing, and (with Sean Eberhard) that they preserve non-trivial association schemes.

#### Integrable groups

With João Araújo, Carlo Casolo and Francesco Matucci, I determined the set of natural numbers n with the property that every group of order n is the derived subgroup of some group. This set is closely related to (but slightly larger than) the set of n for which every group of order n is abelian. We also showed that if a finite group is a derived group, then it is the derived group of a finite group.

• J. Araújo, P. J. Cameron, C. Casolo and F. Matucci, Integrals of groups, Israel J. Math. 234 (2019), 149–178.

#### Equitable partitions of Latin square graphs

With Rosemary Bailey, Alexander Gavrilyuk, and Sergey Goryainov, I determined the equitable partitions of Latin square graphs under an extra assumption on eigenvalues. A partition is equitable if, given any two parts, the number of neighbours in the second part of a vertex in the first depends only on the parts and not on the chosen vertex.

• R. A. Bailey, P. J. Cameron, A. L. Gavrilyuk and S. V. Goryainov, Equitable partitions of Latin square graphs, J. Combinatorial Designs 27 (2019), 142–160.

#### Power graphs of torsion-free groups

In the directed power graph of a group, there is an arc from x to y if y is a power of x; for the undirected power graph, just ignore directions. The power graph does not in general determine the directed power graph, even up to isomorphism (though it does for finite groups). With Horacio Guerra and Šimon Jurina, I showed that for various torsion-free groups, the directions are indeed determined.

• P. J. Cameron, H. Guerra and Š. Jurina, The power graph of a torsion-free group, J. Algebraic Combinatorics 49 (2019), 83-98.

Peter J. Cameron
3 February 2020