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Note on a theorem of Magnus

To Bernhard Hermann Neumann on his 60th birthday

Published online by Cambridge University Press:  09 April 2009

Norman Blackburn
Norman Blackburn
Affiliation:
University of Illinois at Chicago Circle
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Magnus [4] proved the following theorem. Suppose that F is free group and that X is a basis of F. Let R be a normal subgroup of F and write G = F/R. Then there is a monomorphism of F/R′ in which ; here the tx are independent parameters permutable with all elements of G. Later investigations [1, 3] have shown what elements can appear in the south-west corner of these 2 × 2 matrices. In this form the theorem subsequently reappeared in proofs of the cup-product reduction theorem of Eilenberg and MacLane (cf. [7, 8]). In this note a direct group-theoretical proof of the theorems will be given.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 10 , Issue 3-4 , November 1969 , pp. 469 - 474
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Fox, R. H., ‘Free differential calculus, I’, Ann. Math. 57 (1953) 547–560.CrossRefGoogle Scholar
[2]Gaschütz, W., ‘Über modulare Darstellungen endlicher Gruppen, die von freien Gruppen induziert werden’, Math. Z. 60 (1954) 274–286.CrossRefGoogle Scholar
[3]Lyndon, R. C., ‘Cohomology theory of groups with a single defining relation’, Ann. Math. 52 (1950) 650–665.CrossRefGoogle Scholar
[4]Magnus, W., ‘On a theorem of Marshall Hall’, Ann. Math. 40 (1939) 764–768.CrossRefGoogle Scholar
[5]Neumann, B. H., ‘On a theorem of Auslander and Lyndon’, Arch. Math. 13 (1962) 4–9.CrossRefGoogle Scholar
[6]Roquette, P., ‘On class field towers’, Proceedings of an Instructional Conference on algebraic number theory, Thompson, (1967).Google Scholar
[7]Swan, R. G., ‘A simple proof of the cup product reduction theorem’, Proc. Nat. Acad. Sci. 46 (1960) 114–117.CrossRefGoogle ScholarPubMed
[8]Schmid, J., ‘Zu den Reduktionssätzen in der homologischen Theorie der Gruppen’, Arch. Math. 15 (1964) 28–32.CrossRefGoogle Scholar