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On complemented chief factors of finite soluble groups

Published online by Cambridge University Press:  17 April 2009

D.W. Barnes
D.W. Barnes
Affiliation:
Department of Pure Mathematics, University of Sydney, Sydney, New South Wales.
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Abstract

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Let G = H0 > H1 > … > Hr = 1 and G = K0 > K1 > … > Kr =1 be two chief series of the finite soluble group G. Suppose Mi complements Hi/Hi+1. Then Mi also complements precisely one factor Kj/Kj+1, of the second series, and this Kj/Kj+1 is G-isomorphic to Hi/Hi+1. It is shown that complements Mi can be chosen for the complemented factors Hi/Hi+1 of the first series in such a way that distinct Mi complement distinct factors of the second series, thus establishing a one-to-one correspondence between the complemented factors of the two series. It is also shown that there is a one-to-one correspondence between the factors of the two series (but not in general constructible in the above manner), such that corresponding factors are G-isomorphic and have the same number of complements.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 7 , Issue 1 , August 1972 , pp. 101 - 104
Copyright
Copyright © Australian Mathematical Society 1972

References

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