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Keywords:
$n$-ary group; symmetry
$n$-ary group; symmetry
Summary:
Properties of $n$-ary groups connected with the affine geometry are considered. Some conditions for an $n$-ary $rs$-group to be derived from a binary group are given. Necessary and sufficient conditions for an $n$-ary group $<\theta ,b>$-derived from an additive group of a field to be an $rs$-group are obtained. The existence of non-commutative $n$-ary $rs$-groups which are not derived from any group of arity $m<n$ for every $n\ge 3$, $r>2$ is proved.
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