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Note on p-Groups

Published online by Cambridge University Press:  22 January 2016

Noboru Itô
Noboru Itô*
Affiliation:
Nagoya University
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In connection with the class field theory a problem concerning p-groups was proposed by W. Magnus: Is there any infinite tower of p-groups G1, G2,…, Gn, Gn+1,…such that G1 is abelian and each Gn is isomorphic to Gn+1/θn(Gn+1), θn(Gn+1) ≠ 1, n = 1,2,…, where θn(Gn+1) denotes the n-th commutator subgroup of Gn+1? The present note is, firstly, to construct indeed such a tower, to settle the problem, and also to refine an inequality for p-groups of P. Hall.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 1 , June 1950 , pp. 113 - 116
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1950

References

1) Magnus, W., Beziehung zwishen Gruppen und Idealen in einem speziellen Ring, Math. Annalen 111 (1935).CrossRefGoogle Scholar

2) An impulse was given to the present work by Dr. K. Iwasawa, through a communication by Mr. M. Suzuki.

3) Hall, P., A contribution to the theory of groups of prime power order, Proc. London Math. Soc. 36 (1934).Google Scholar