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On Associative Compositions in Finite Nilpotent Groups
Published online by Cambridge University Press: 22 January 2016
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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
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- 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.
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