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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

1. Cohen, I. S., On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), 54–106.Google Scholar
2. Remeslennikov, V. N., Finite approximability of metabelian groups (Russian), Alg. i Logika 7 (1968), 106-113; Alg. and Logic 7 (1968), 268–272.Google Scholar
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
5. Zariski, O. and Samuel, P., Commutative algebra Vol. 1 (Van Nostrand-Princeton, 1958).Google Scholar