Let $\cdot$, $\varphi$ and $b$ be a binary operation in $Q$ , a unary operation in $Q$, and a constant in $Q$, respectively. Let, also, $n\in N\backslash\{1,2\}$. Then, in the present article, the algebra $(Q,\{\cdot,\varphi,b\})$ is said to be a Hosszú-Glushkin algebra of order $n$ (briefly: $n$HG-algebra) iff the following hold: 1. $(Q,\cdot)$ is a group; 2. $\varphi\in Aut(Q,\cdot)$; 3. $\varphi(b)=b$; and 4. $\varphi^{n-1}(x)\cdot b=b\cdot x$ for every $x\in Q$. Under this condition the Hosszú-Glushkin Theorem [2, 3] can be formulated in the following way: If $(Q,A)$ is an $n$-group and $n\in N\backslash\{1,2\}$, then there is an $n$HG-algebra $(Q,\{\cdot,\varphi,b\})$ such that $A(x_1,\dots,x_n)=x_1\cdot\varphi(x_2)\cdot\dots\cdot\varphi^{n-2}(x_{n-1})\cdot b\cdot x_n$ for every $x_1,\dots,x_n\in Q$. Then, we say that this $n$HG-algebra is a \emph{corresponding} $n$HG-algebra for the $n$-group $(Q,A)$. The main result of the paper is a description of all $n$HG-algebras corresponding to an $n$-group $(Q,A)$, by means of one of them (Theorem 5.1).