Solution of Belousov's problem

Details Solution of Belousov's problem Solution of Belousov's problem

2000-10-17

arxiv.org/abs/math/0010175v1

Discuss. Math. Gen. Algebra Appl. 51 (2001) no. 1 193-203

Mathematics

Group Theory

AMS-LaTeX, 7 pages

Scientific paper

The authors prove that a local $n$-quasigroup defined by the equation x_{n+1} = F (x_1, ..., x_n) = [f_1 (x_1) + ... + f_n (x_n)]/[x_1 + ... + x_n], where f_i (x_i), i, j = 1, ..., n, are arbitrary functions, is irreducible if and only if any two functions f_i (x_i) and f_j (x_j), i \neq j, are not both linear homogeneous, or these functions are linear homogeneous but f_i (x_i)/x_i \neq f_j (x_j)/x_j. This gives a solution of Belousov's problem to construct examples of irreducible $n$-quasigroups for any n \geq 3.

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