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Faithful Representations of Finitely Generated Metabelian Groups

Published online by Cambridge University Press:  20 November 2018

B. A. F. Wehrfritz
B. A. F. Wehrfritz*
Affiliation:
Queen Mary College, London, England
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In [3] Remeslennikov proves that a finitely generated metabelian group G has a faithful representation of finite degree over some field F of characteristic zero (respectively, p > 0) if its derived group G’ is torsion-free (respectively, of exponent p). By the Lie-Kolchin-Mal'cev theorem any metabelian subgroup of GL(n, F) has a subgroup of finite index whose derived group is torsion-free if char F = 0 and is a p-group of finite exponent if char F = p > 0. Moreover every finite extension of a group with a faithful representation (of finite degree) has a faithful representation over the same field. Thus Remeslennikov's results have a gap which we propose here to fill.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 6 , 01 December 1975 , pp. 1355 - 1360
Copyright
Copyright © Canadian Mathematical Society 1975

References

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3. Remeslennikov, V. N., Representations of finitely generated metabelian groups by matrices (Russian), Alg. i Logika 8 (1969), 72-75; Alg. and Logic 8 (1969), 39–40 Google Scholar
4. Wehrfritz, B. A. F., Infinite linear groups, Ergeb. d. Math. Bd. 76 (Springer-Berlin Heidelberg New York, 1973).Google Scholar
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