The main result of the article is the following proposition. Let $n\geq 2m$ and let $(Q,A)$ be an $(n,m)$-groupoid. Then, $(Q,A)$ is an $(n,m)$-group iff the following statements hold: $(i)$ $(Q,A)$ is an $\langle 1,n-m+1\rangle$- and $\langle 1,2\rangle$-associative $(n,m)$-groupoid [or $\langle 1,n-m+1\rangle$- and $\langle n-m,n-m+1\rangle$-associative $(n,m)$-groupoid]; and $(ii)$ for every $a_{1}^{n}\in Q$ there is <b>at least one</b> $x_{1}^{m}\in Q^{m}$ and <b>at least one</b> $y_{1}^{m}\in Q^{m}$ such that the following equalities hold $A(a_{1}^{n-m},x_{1}^{m}) = a_{n-m+1}^{n}$ and $A(y_{1}^{m},a_{1}^{n-m})=a_{n-m+1}^{n}$. [For $n=2$ and $m=1$ it is a well known characterization of a group. See, also 3.2]