Schreier generators I am facing some problem in understanding the proof of the following theorem. Can somebody provide me a simple proof .
Given $G=\langle A \rangle$ and $H \leq G$, and $R$ is the coset representatives for $H$ in $G$.
Let $B=\{r_1ar^{-1}_2 | r_1,r_2 \in R, a \in A\}\cap H.$ 
Then $B$ generates $H$.

Your assertion is a few incorrect. It must be as follows:
Given $G=\langle A \rangle$ and $H\le G$, and $R$ is a set of representatives  of the  right cosets for H in G. Let B={r1ar−12|r1,r2∈R,a∈A}∩H. Then B generates H. Let     $B = \{r_1ar^{-1}_2 | r_1 \in R, a \in A\}$, where $r_2$ is the representative of the coset $Hr_1a$. Then $B$ generates $H$.
You can find a proof in:
M.I. Kargapolov, Ju.I. Merzljakov, Fundamentals of the Theory of Groups, Springer, Graduate Texts in Math (62), Theorem 14.3.1. 
