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Enumerating isoclinism classes of semi-extraspecial groups

Part of: Representation theory of groups

Published online by Cambridge University Press:  24 February 2020

Mark L. Lewis  and
Joshua Maglione
Mark L. Lewis
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, OH44242, USA ([email protected])
Joshua Maglione
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, D-33501Bielefeld, Germany ([email protected])
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Abstract

We enumerate the number of isoclinism classes of semi-extraspecial p-groups with derived subgroup of order p2. To do this, we enumerate GL (2, p)-orbits of sets of irreducible, monic polynomials in 𝔽p[x]. Along the way, we also provide a new construction of an infinite family of semi-extraspecial groups as central quotients of Heisenberg groups over local algebras.

Keywords

Pfaffiangenus 2bilinear maps

MSC classification

Primary: 20D15: Nilpotent groups, $p$-groups
Secondary: 20D60: Arithmetic and combinatorial problems
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 2 , May 2020 , pp. 426 - 442
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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References

1.Artin, E., Geometric algebra (Interscience Publishers, New York–London, 1957).Google Scholar
2.Beisiegel, B., Semi-extraspezielle p-Gruppen, Math. Z. 156(3) (1977), 247–254 (in German).CrossRefGoogle Scholar
3.Brooksbank, P. A., Maglione, J. and Wilson, J. B., A fast isomorphism test for groups whose Lie algebra has genus 2, J. Algebra 473 (2017), 545–590.CrossRefGoogle Scholar
4.Brooksbank, P. A., O'Brien, E. A. and Wilson, J. B., Testing isomorphism of graded algebras, Trans. Amer. Math. Soc. 372(11) (2019), 8067–8090.CrossRefGoogle Scholar
5.Camina, A. R., Some conditions which almost characterize Frobenius groups, Israel J. Math. 31(2) (1978), 153–160.CrossRefGoogle Scholar
6.Chillag, D., Mann, A. and Scoppola, C. M., Generalized Frobenius groups. II, Israel J. Math. 62(3) (1988), 269–282.CrossRefGoogle Scholar
7.Dempwolff, U., Semifield planes of order 81, J. Geom. 89(1–2) (2008), 1–16.CrossRefGoogle Scholar
8.Dickson, L. E., Linear groups: with an exposition of the Galois field theory, with an introduction by W. Magnus, (Dover Publications, New York, 1958).Google Scholar
9.Dieudonné, J., Sur la réduction canonique des couples de matrices, Bull. Soc. Math. France 74 (1946), 130–146 (in French).CrossRefGoogle Scholar
10.Dornhoff, L., Group representation theory. Part A: Ordinary representation theory, Pure and Applied Mathematics, Volume 7 (Marcel Dekker, New York, 1971).Google Scholar
11.Hall, P., The classification of prime-power groups, J. Reine Angew. Math. 182 (1940), 130–141.Google Scholar
12.Huppert, B., Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134 (Springer-Verlag, Berlin–New York, 1967) (in German).CrossRefGoogle Scholar
13.Johnson, N. L., Jha, V. and Biliotti, M., Handbook of finite translation planes, Pure and Applied Mathematics (Boca Raton), Volume 289 (Chapman & Hall/CRC, Boca Raton, FL, 2007).CrossRefGoogle Scholar
14.Kantor, W. M., Finite semifields, Finite Geometries, Groups, and Computation, pp. 103–114 (Walter de Gruyter, Berlin, 2006).Google Scholar
15.King, O. H., The subgroup structure of finite classical groups in terms of geometric configurations, in Surveys in Combinatorics 2005, London Mathematical Society Lecture Note Series, Volume 327, pp. 29–56 (Cambridge University Press, Cambridge, 2005).CrossRefGoogle Scholar
16.Lewis, M. L., Semi-extraspecial groups with an abelian subgroup of maximal possible order, Adv. Group Theory Appl. 5 (2018), 91–122.Google Scholar
17.Lewis, M. L., Semi-extraspecial groups, in Advances in algebra, Springer Proceedings in Mathematics and Statistics, Volume 277, pp. 219–237 (Springer, Cham, 2019).CrossRefGoogle Scholar
18.Lewis, M. L. and Wilson, J. B., Isomorphism in expanding families of indistinguishable groups, Groups Complex. Cryptol. 4(1) (2012), 73–110.CrossRefGoogle Scholar
19.Macdonald, I. D., Some p-groups of Frobenius and extra-special type, Israel J. Math. 40(3–4) (1981), 350–364 (1982).CrossRefGoogle Scholar
20.Maglione, J., Efficient characteristic refinements for finite groups. Part 2, J. Symbolic Comput. 80(part 2) (2017), 511–520.CrossRefGoogle Scholar
21.Scharlau, R., Paare alternierender Formen, Math. Z. 147(1) (1976), 13–19.CrossRefGoogle Scholar
22.Steinberg, R., The representations of GL(3, q), GL(4, q), PGL(3, q), and PGL(4, q), Canadian J. Math. 3 (1951), 225–235.CrossRefGoogle Scholar
23.Vaughan-Lee, M., Orbits of irreducible binary forms over GF(p), preprint. (arXiv:1705.07418, 2017).Google Scholar
24.Verardi, L., Semi-extraspecial groups of exponent p, Ann. Mat. Pura Appl. (4) 148 (1987), 131–171 (in Italian, with English summary).CrossRefGoogle Scholar
25.Vishnevetskiĭ, A. L., The number of classes of projectively equivalent binary forms over a finite field, Dokl. Akad. Nauk Ukrain. SSR Ser. A 4 (1982), 9–12, 86 (in Russian, with English summary).Google Scholar
26.Wilson, J. B., Existence, algorithms, and asymptotics of direct product decompositions, I, Groups Complex. Cryptol. 4(1) (2012), 33–72.CrossRefGoogle Scholar
27.Wilson, J. B., More characteristic subgroups, Lie rings, and isomorphism tests for p-groups, J. Group Theory 16(6) (2013), 875–897.CrossRefGoogle Scholar