A part, of [2, Theorem 1.4] is the following Proposition: Let $n\geq3$ and let $(Q,A)$ be an $n$-semigroup (1.2). Then $(Q,A)$ is an $n$-group (1.2) iff for arbitrary $i\in\{2,\dots,n-1\}$ for every $a^n_1\in Q$ (1.1) there is exactly one $x\in Q$ such that the following equality holds $A(a^{i-1}_1,x,a^{n-1}_i)=a_n$ (1.1). In the present paper the following proposition is proved: Let $n\geq3$ and let $(Q,A)$ be an $n$-groupoid. Then $(Q,A)$ is an $n$-group iff for an arbitrary $i\in\{2,\dots,n-1\}$ the following condition hold: (a) the $\langle i-1,i\rangle$-associative law holds in $(Q,A)$; (b) the $\langle i,i+1\rangle$-associative law holds in $(Q,A)$; and (c) for every $a^n_1\in Q$ there is exactly one $x\in Q$ such that the following equality holds $A(a^{i-1}_1,x,a^{n-1}_i)=a_n$. In addition, for $n=3$ $[i=2]$ the conditions (a) and (b) are equivalent to the condition that $(Q,A)$ is a 3-semigroup (1.2).