Classification of the Finite N-Generator Transvection Groups Over Z<sub><small>2</small></sub>

Advances in Applied Mathematics Vol. 44 Issue 3 (March 2010) 185–202

Classification of the finite n-generator transvection groups over Z₂^✩

Jizhu Nan*, Jing Zhao

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China

Article history:
Received 29 October 2008
Revised 5 June 2009
Accepted 3 July 2009
Available online 17 September 2009

MSC:
15A33
51F15

Keywords:
Transvection
Effective irreducible reflection group
Cartan matrix

Abstract

We study and give a complete classification of the finite n-generator
transvection groups over Z₂.

Introduction

Problems concerning the classification of finite reflection groups are traditional in the theory of
finite groups. Reflection groups over real Euclidean spaces are treated in, for example, references
[3,4,6,12]. G.C. Shephard and J.A. Todd classified the complex reflection groups in reference [11] but
A.M. Cohen gave an independent treatment in reference [1] and he also discussed quaternionic reflection
groups in reference [2]. The papers [13–15] enumerate all finite irreducible reflection groups
over a field of finite characteristic p > 2. The papers [8–10] study the reflection groups of references
[13–15] in the context of regular polytopes over finite fields. Irreducible groups of linear
transformations generated by transvections over Z₂, the integers module 2, have been classified by
J.E. McLaughlin in reference [7], by mainly using geometrical methods. In the present paper, by using
the matrix method, we succeed in describing the irreducible transvection groups in terms of generators
and relations, and up to conjugacy in the general linear group GL_m(Z₂), we give a complete
classification of the finite n-generator transvection groups over Z₂. In the end of this paper, as an
example, we indicate how to find explicit generators of all the irreducible n-generator transvection
groups.

✩ Supported by the National Natural Science Foundation of China (grant No. 10771023).
* Corresponding author.

References

[1] A.M. Cohen, Finite complex reflection groups, Ann. Sci. Ecole Norm. Sup. 9 (1976) 379–436.
[2] A.M. Cohen, Finite quaternionic reflection groups, J. Algebra 64 (1980) 293–324.
[3] H.S.M. Coxeter, Discrete groups generated by reflections, Canad. J. Math. 35 (1934) 588–621.
[4] L.C. Grove, C.T. Benson, Finite Reflection Groups, Springer-Verlag, New York, 1971.
[5] L.K. Hua, Z.X. Wan, Classical Groups, Shanghai Science and Technology Press, Shanghai, 1963 (in Chinese).
[6] J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Univ. Press, Cambridge, 1990.
[7] J.E. McLaughlin, Some subgroups of SLn(F2), Illinois J. Math. 13 (1969) 108–115.
[8] B. Monson, E. Schulte, Reflection groups and polytopes over finite fields I, Adv. in Appl. Math. 33 (2004) 290–317.
[9] B. Monson, E. Schulte, Reflection groups and polytopes over finite fields II, Adv. in Appl. Math. 38 (2007) 327–356.
[10] B. Monson, E. Schulte, Reflection groups and polytopes over finite fields III, Adv. in Appl. Math. 41 (2008) 76–94.
[11] G.C. Shephard, J.A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954) 274–304.
[12] R. Steinberg, Finite reflection groups, Trans. Amer. Math. Soc. 91 (1959) 493–504.
[13] A. Wagner, Determination of the finite primitive reflection groups over an arbitrary field of characteristic not two I, II, III,
Geom. Dedicata 9 (1980) 239–253, Geom. Dedicata 10 (1981) 191–203, Geom. Dedicata 10 (1981) 475–523.
[14] A.E. Zalesski˘ı, V.N. Serežkin, Linear groups generated by transvections, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976) 26–49
(in Russian); English translation in Math. USSR-Izv. 10 (1976) 25–46.
[15] A.E. Zalesski˘ı, V.N. Serežkin, Finite linear groups generated by reflections, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980) 1279–
1307 (in Russian); English translation in Math. USSR-Izv. 17 (1981) 477–503.