Consider a presentation $\mathcal{ P}=<{ \bf x}\mid{ \bf \bigcup_{ i=1}^n r_i}>$. Let ${ \bf R_i}$ be the normal closure of the set ${ \bf r_i}$ in the free group ${ \bf F}$ with basis ${ \bf x}$, $\mathcal{ P}_i=<{ \bf x}\mid{ \bf r_i}>$, ${ \bf N_i} = \prod_{ j\neq i}{ \bf R_j}$. In the present article, using geometric techniques of pictures, generators for $\frac{ { \bf R_i}\cap { \bf N_i}}{ [{ \bf R_i}, { \bf N_i}]}$, $i=1,...,n$, are obtained from a set of generators over $\{ \mathcal{ P}_i\mid i=1,..., n\}$ for $\pi_2(\mathcal{ P})$. As a corollary, we get a sufficient condition for the family $\{ { \bf R_1},...,{ \bf R_n}\}$ to be independent.
Categories: math.GR