Hostname: page-component-cb9f654ff-fg9bn Total loading time: 0 Render date: 2025-09-02T06:52:18.008Z Has data issue: false hasContentIssue false
  • English
  • Français

ON 2-TRANSITIVE SETS OF EQUIANGULAR LINES

Part of: Discrete geometry Permutation groups Graph theory

Published online by Cambridge University Press:  22 August 2022

ULRICH DEMPWOLFF Open the ORCID record for ULRICH DEMPWOLFF [Opens in a new window]  and
WILLIAM M. KANTOR Open the ORCID record for WILLIAM M. KANTOR [Opens in a new window]
ULRICH DEMPWOLFF
Affiliation:
Department of Mathematics, University of Kaiserslautern, Kaiserslautern 67653, Germany e-mail: dempwolff@mathematik.uni-kl.de
WILLIAM M. KANTOR*
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
*
e-mail: kantor@uoregon.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We determine all finite sets of equiangular lines spanning finite-dimensional complex unitary spaces for which the action on the lines of the set-stabiliser in the unitary group is 2-transitive with a regular normal subgroup.

Keywords

2-transitiveequiangular lines

MSC classification

Primary: 52C35: Arrangements of points, flats, hyperplanes
Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures 81P15: Quantum measurement theory

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 107 , Issue 1 , February 2023 , pp. 134 - 145
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Aschbacher, M., Finite Group Theory, 2nd edn (Cambridge University Press, Cambridge 2000).10.1017/CBO9781139175319CrossRefGoogle Scholar
Bolt, B., Room, T. G. and Wall, G. E., ‘On the Clifford collineation, transform and similarity groups I, II’, J. Aust. Math. Soc. 2 (1961), 60–79, 80–96.10.1017/S1446788700026379CrossRefGoogle Scholar
Cameron, P. J., ‘Primitive permutation groups and finite simple groups’, Bull. Lond. Math. Soc. 13 (1981), 1–22.10.1112/blms/13.1.1CrossRefGoogle Scholar
Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A. and Thackray, J. G., Atlas of Finite Groups (Oxford University Press, Eynsham 1985).Google Scholar
Dickson, L. E., ‘The groups of Steiner in problems of contact (second paper)’, Trans. Amer. Math. Soc. 3 (1902), 377–382.Google Scholar
Gérardin, P., ‘Weil representations associated to finite fields’, J. Algebra 46 (1977), 54–101.10.1016/0021-8693(77)90394-5CrossRefGoogle Scholar
Griess, R. L. Jr, ‘Automorphisms of extra special groups and nonvanishing degree $2$ cohomology’, Pacific J. Math. 48 (1973), 403–422.10.2140/pjm.1973.48.403CrossRefGoogle Scholar
Hoggar, S. G., ‘64 lines from a quaternionic polytope’, Geom. Dedicata 69 (1998), 287–289.10.1023/A:1005009727232CrossRefGoogle Scholar
Huppert, B., ‘Singer–Zyklen in klassischen Gruppen’, Mat. Z. 117 (1970), 141–150.10.1007/BF01109836CrossRefGoogle Scholar
Huppert, B. and Blackburn, N., Endliche Gruppen I (Springer, Berlin, 1967), Finite Groups II, III (Springer, Berlin, 1982).10.1007/978-3-642-64981-3CrossRefGoogle Scholar
Iverson, J. W. and Mixon, D. G., ‘Doubly transitive lines II: almost simple symmetries’, Preprint, 2019, arXiv:1905.06859.Google Scholar
Jones, W. and Parshall, B., ‘On the 1-cohomology of finite groups of Lie type’, Proc. Conf. Finite Groups (Utah 1975) (eds. Scott, W. R. and Gross, F.) (Academic Press, New York, 1976), 313–328.10.1016/B978-0-12-633650-4.50022-9CrossRefGoogle Scholar
Landazuri, V. and Seitz, G. M., ‘On the minimal degrees of projective representations of the finite Chevalley groups’, J. Algebra 32 (1974), 418–443.10.1016/0021-8693(74)90150-1CrossRefGoogle Scholar
Lemmens, P. E. H. and Seidel, J. J., ‘Equiangular lines’, J. Algebra 24 (1973), 494–512.10.1016/0021-8693(73)90123-3CrossRefGoogle Scholar
Liebeck, M. W., ‘The affine permutation groups of rank three’, Proc. Lond. Math. Soc. (3) 54 (1987), 477–516.10.1112/plms/s3-54.3.477CrossRefGoogle Scholar
Renes, J. M., Blume-Kohout, R., Scott, A. J. and Caves, C. M., ‘Symmetric informationally complete quantum measurements’, J. Math. Phys. 45 (2004), 2171–2180.10.1063/1.1737053CrossRefGoogle Scholar
Taylor, D. E., The Geometry of the Classical Groups (Heldermann, Berlin, 1992).Google Scholar
Taylor, D. E., ‘Two-graphs and doubly transitive groups’, J. Combin. Theory Ser. A 61 (1992), 113–122.10.1016/0097-3165(92)90056-ZCrossRefGoogle Scholar
Thompson, J., ‘Nonsolvable finite groups all of whose local subgroups are solvable’, Bull. Amer. Math. Soc. (N.S.) 74 (1968), 383–437.10.1090/S0002-9904-1968-11953-6CrossRefGoogle Scholar
Tiep, P. H. and Zaleskii, A. E., ‘Some aspects of finite linear groups: a survey’, J. Math. Sci. (N.Y.) 100 (2000), 1893–1914.10.1007/BF02677502CrossRefGoogle Scholar
Winter, D., ‘The automorphism group of an extraspecial $p$ -group’, Rocky Mountain J. Math. 2 (1972), 159–168.10.1216/RMJ-1972-2-2-159CrossRefGoogle Scholar
Zhu, H., ‘Super-symmetric informationally complete measurements’, Ann. Physics 362 (2015), 311–326.10.1016/j.aop.2015.08.005CrossRefGoogle Scholar