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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The geometry of diagonal groups
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by R. A. Bailey, Peter J. Cameron, Cheryl E. Praeger and Csaba Schneider PDF
Trans. Amer. Math. Soc. 375 (2022), 5259-5311 Request permission


Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O’Nan–Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have not been studied in general.

The main purpose of this paper is to describe and characterise such structures, which we call diagonal semilattices. Unlike the diagonal groups in the O’Nan–Scott theorem, which are defined over finite characteristically simple groups, our construction works over arbitrary groups, finite or infinite.

A diagonal semilattice depends on a dimension $m$ and a group $T$. For $m=2$, it is a Latin square, the Cayley table of $T$, though in fact any Latin square satisfies our combinatorial axioms. However, for $m \geqslant3$, the group $T$ emerges naturally and uniquely from the axioms. (The situation somewhat resembles projective geometry, where projective planes exist in great profusion but higher-dimensional structures are coordinatised by an algebraic object, a division ring.)

A diagonal semilattice is contained in the partition lattice on a set $\Omega$, and we provide an introduction to the calculus of partitions. Many of the concepts and constructions come from experimental design in statistics.

We also determine when a diagonal group can be primitive, or quasiprimitive (these conditions turn out to be equivalent for diagonal groups).

Associated with the diagonal semilattice is a graph, the diagonal graph, which has the same automorphism group as the diagonal semilattice except in four small cases with $m\leqslant 3$. The class of diagonal graphs includes some well-known families, Latin-square graphs and folded cubes, and is potentially of interest. We obtain partial results on the chromatic number of a diagonal graph, and mention an application to the synchronization property of permutation groups.

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Additional Information
  • R. A. Bailey
  • Affiliation: School of Mathematics and Statistics, University of St Andrews, St Andrews, Fife KY16 9SS, United Kingdom
  • ORCID: 0000-0002-8990-2099
  • Email:
  • Peter J. Cameron
  • Affiliation: School of Mathematics and Statistics, University of St Andrews, St Andrews, Fife KY16 9SS, United Kingdom
  • MR Author ID: 44560
  • ORCID: 0000-0001-9699-4836
  • Email:
  • Cheryl E. Praeger
  • Affiliation: Department of Mathematics and Statistics, University of Western Australia, Perth, WA 6009, Australia
  • MR Author ID: 141715
  • ORCID: 0000-0002-0881-7336
  • Email:
  • Csaba Schneider
  • Affiliation: Departimento di Matemática, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil
  • MR Author ID: 624450
  • Email:
  • Received by editor(s): August 1, 2020
  • Received by editor(s) in revised form: May 4, 2021
  • Published electronically: May 23, 2022
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 5259-5311
  • MSC (2020): Primary 20B05; Secondary 20B07, 20B15, 05B15, 62K15
  • DOI:
  • MathSciNet review: 4469221