Group is n-ary group for all n

Statement

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 $n\geq 2$. 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:

$f(a_{1},a_{2},\dots ,a_{n})=a_{1}a_{2}\dots a_{n}$