Representations of polycyclic groups

Richard G. Swan

Type: Article

Publication Date: 1967-06-01

Citations: 37

DOI: https://doi.org/10.1090/s0002-9939-1967-0213442-5

Abstract

L. Auslander [1] has recently shown that every polycyclic group' has a faithful representation in GL(n, Z) for some n, thus solving a problem of P. Hall [2]. His proof involves considerable knowledge of the theory of Lie groups. Since the result obtained is purely algebraic, it is of interest to find a purely algebraic proof of it. It struck me that the proof of Ado's theorem [5] could be adapted to this purpose and I will show here that this is indeed the case. I would like to thank J. Thompson and J. Alperin for calling this problem to my attention. Recall that a matrix (aij) is called uni-triangular if aij = 0 for j <i and aii = i for all i. These form a nilpotent subgroup Tn(Z) of GL(n, Z).

Locations

Proceedings of the American Mathematical Society - View - PDF

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