Given positive integer $n$, an element $\alpha$ of a field is an nth root of unity if $\alpha^n=1$. It is a primitive nth root of unity if $\alpha^n=1$ and $\alpha^m\neq1$ for $0<m<n$.
I've seen this definition of primitive roots of unity for fields, and even for rings with unity, but not for groups. Would the definition make sense, and if not, why not?
Remark I suppose if this definition makes sense, the torsion subgroup would precisely contain the nth roots of unity for each $n$?