Here is the problem:
Let $G=\langle x_0,x_1,x_2,\ldots\ |px_0=0,x_0=p^nx_n, \text{all } n\geq1\rangle$. Prove that $G/\langle x_0\rangle$ is a direct sum of cyclic groups and is reduced.
The first part is easy because if we put the relation $x_0=0$ to other relations in $G$; $G$ would be a direct sum of cyclic groups. For another part, I feel that the first part is usefull, but I don't know how to link these together. The following ideas just came to me:
- if I assume it is not reduced; $dG$ would be a proper subgroup and so $G/dG$ is reduced.
or
- To show that $\{0\}$ is the only divisible subgroup of $G$.
Thanks for your time.