Geodesic
Two Results on Associativity of Composite Operations in Groups
Publications de l'Institut Mathématique, _N_S_33 (1983) no. 47, p. 123

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The theorem of Hanna Neumann ([{\bf 4}]) states that all associative operations $w(x,y)$ in the case of a free $G$ are of one of the following forms: $ a, x, y, xay, yax, $ where a is an arbitrary element of $G$. In the first part of this article we generalize this result. Theorem 1 shows that operations of the forms listed above are the only possible (except trivial cases) when we require $w(x,y)$ to satisfy not the associativity law, but any consequence of it (any weakened associativity law). In the second part of the article we determine all associative operations $w(x,y)$ in the case of $G$ free nilpotent of class two.
Classification : 20A99 
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     author = {Sava Krsti\'c},
     title = {Two {Results} on {Associativity} of {Composite} {Operations} in {Groups}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {123 },
     publisher = {mathdoc},
     volume = {_N_S_33},
     number = {47},
     year = {1983},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_1983_N_S_33_47_a17/}
}
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Sava Krstić. Two Results on Associativity of Composite Operations in Groups. Publications de l'Institut Mathématique, _N_S_33 (1983) no. 47, p. 123 . http://geodesic.mathdoc.fr/item/PIM_1983_N_S_33_47_a17/