# Group is n-ary group for all n

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Suppose $G$ is a group where we denote the multiplication by concatenation (i.e., we omit the multiplication symbol). Let $n$ be an integer with $n \ge 2$. We can equip $G$ with the structure of a $n$-ary group (i.e., a multiary group for arity $n$) as follows: we define the $n$-ary operation $f:G^n \to G$ as:
$f(a_1,a_2,\dots,a_n) = a_1a_2 \dots a_n$