Czechoslovak Mathematical Journal, Vol. 67, No. 3, pp. 655-698, 2017
Invariants of finite groups generated by generalized transvections in the modular case
Xiang Han, Jizhu Nan, Chander K. Gupta
Received February 2, 2016. First published July 12, 2017.
Abstract: We investigate the invariant rings of two classes of finite groups $G\leq{\rm GL}(n,F_q)$ which are generated by a number of generalized transvections with an invariant subspace $H$ over a finite field $F_q$ in the modular case. We name these groups generalized transvection groups. One class is concerned with a given invariant subspace which involves roots of unity. Constructing quotient groups and tensors, we deduce the invariant rings and study their Cohen-Macaulay and Gorenstein properties. The other is concerned with different invariant subspaces which have the same dimension. We provide a explicit classification of these groups and calculate their invariant rings.
Keywords: invariant ring; transvection; generalized transvection group
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Affiliations: Xiang Han, Jizhu Nan, School of Mathematical Sciences, Dalian University of Technology, No. 2 Linggong Road, Dalian, 116024, Ganjingzi, Liaoning, P. R. China, e-mail: xianghan328@yahoo.com, jznan@163.com; Chander K. Gupta, Department of Mathematics, University of Manitoba, Machray Hall 420, 186 Dysart Rd, Winnipeg, MB R3T 2M8, Canada
