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On Associative Compositions in Finite Nilpotent Groups

Published online by Cambridge University Press:  22 January 2016

Nagayosi Iwahori  and
Akira Hattori
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Let

(1)

be a word in two variables X, Y, i.e. an element in the free group F2 on two generators X, Y. Let us say that f defines an associative composition for a group G if for arbitrary elements a, b, c in G we have

(2)

where a ° b is defined by

(3)

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 7 , June 1954 , pp. 145 - 148
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1954

References

1 Iwasawa, K., Einige Satze uber freie Gruppen, Proc. Imp. Acad. Japan, 19 (1943), pp. 272–274 Google Scholar.

2 Note that is a central subgroup of Then for every a, b in G, c, d in we have (cf. H. Zassenhaus, Lehrbuch der Gruppentheorie, S. 57). The assertion is then completed by induction on i.