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The Fitting subgroup of a linear solvable group

Published online by Cambridge University Press:  09 April 2009

John D. Dixon
John D. Dixon
Affiliation:
University of New South WalesKensington, N.S.W.
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Let G be a group. The Fitting subgroup F(G) of G is defined to be the set union of all normal nilpotent subgroups of G. Since the product of two normal nilpotent subgroups is again a normal nilpotent subgroup (see [10] p. 238), F(G) is the unique maximal normal, locally nilpotent sungroup of G. In particular, is G is finite, then F(G) is the unique maximal normal nilpotent subgroup of G. If G is a notrivial solvable group, then clearly F(G) ≠1.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 7 , Issue 4 , November 1967 , pp. 417 - 424
Copyright
Copyright © Australian Mathematical Society 1967

References

[1]Artin, E., Geometric algebra (Interscience 1957).Google Scholar
[2]Bercov, R. and Moser, L., ‘On Abelian permutation groups’, Canad. Math. Bull. 8 (1965), 627–630.CrossRefGoogle Scholar
[3]Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (Interscience 1962).Google Scholar
[4]Dickson, L. E., Linear groups (reprint), (Dover 1958).Google Scholar
[5]Dixon, J. D., ‘Complete reducibility of infinite groups’, Canad. J. Math. 16 (1964), 267–274.CrossRefGoogle Scholar
[6]Huppert, B., ‘Lineare auflösbare Gruppen’, Math. Z. 67 (1957), 479–518.CrossRefGoogle Scholar
[7]Scott, W. R., Theory of groups (Prentice-Hall 1964).Google Scholar
[8]Suprunenko, D., ‘Soluble and nilpotent linear groups’, Amer. Math. Soc. Transl. Math. Monographs vol. 9, (1963).Google Scholar
[9]Tschebotaröw, N. and Schwerdtfeger, H., Grundzüge der Galois'schen Theorie, (Noordhoff 1950).Google Scholar
[10]Zassenhaus, H., Theory of groups (2nd ed., Chelsea 1958).Google Scholar