that for each $P \in \mathcal{R}_{m}$, there is $\beta_P \in \T(X,Y)$ with $\ker \beta_P = P$, which will be fixed. Let $\mathcal{B} := \{\beta_P : P \in \mathcal{R}_{m}\}$. If $m=2$ then $\mathcal{R}_{m}=\emptyset$ and $\S(Y)=\{\eta |_{Y},(\eta |_{Y})^{2}\} = \langle \eta |_{Y} \rangle$, obviously. If $m\geq 3$ then without loss of generality we can assume that there is $P'\in \mathcal{R}_{m}$ such that $Y$ is a transversal of $\ker \beta_{P'}$ and $\beta_{P'}|_{Y}=\left(
\begin{array}{ccccc}
a_{1} & a_{2} & a_{3} & \cdots  & a_{m} \\
a_{2} & a_{1} & a_{3} & \cdots  & a_{m}\end{array}\right)$. It is well known that $\S(Y)=\left\langle \beta_{P'}|_Y, \eta|_Y \right\rangle$. Hence, $\mathcal{B}$ is a relative generating set of $\T(X,Y)$ modulo $\OP(X,Y)$ by Lemma \[le7\]. Since $|\mathcal{B}| = |\mathcal{R}_{m}|$, we obtain the required result.

Now we will characterize the minimal relative generating sets of $\T(X,Y)$ modulo $\OP(X,Y)$. The minimal relative generating sets do not coincide with the relative generating sets of size $\rank(\T(X,Y):\OP(X,Y))$.

Let $B\subseteq \T(X,Y)$. Then $B$ is a minimal relative generating set of $\T(X,Y)$ modulo $\OP(X,Y)$ if and only if there is a set $\widetilde{B}\subseteq B$ such that the following three statements are satisfied:$(i)$ $\mathcal{R}_{m} \subseteq \{\ker \beta :\beta \in B\setminus \widetilde{B}\}$,$(ii)$ $|B\setminus \widetilde{B}| = |\mathcal{R}_{m}|$, $(iii)$ $\S(Y) \subseteq \left\langle \{\beta|_{Y}:\beta 