Prove that U_{m/n_1} (m) ,U_{m/n_k} (m) are normal subgroups

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prove that U_{m/n_1} (m) , ... U_{m/n_k} (m) are normal subgroups

In the attached image I have proved that U_{m/n_1} (m) , ... U_{m/n_k} (m) are normal subgroups

But how do i Prove that U(m) = U_{m/n_1} (m) ... U_{m/n_k} (m)?

and that their intersection is identity alone.

Help will be appreciated. Thanks
 

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