Originally Posted by

**deniselim17** I'm stuck at this question for more than 24 hours. This question is from "THE THEORY OF GROUPS: An Introduction" by Joseph J. ROTMAN, Exercise 4.11.

Here is the question:

If $\displaystyle G$ is a group with normal subgroups $\displaystyle H_{1},H_{2}, ... ,H_{m} $, then $\displaystyle G=\prod_{i=1}^{m} H_{i}$(internal) if and only if $\displaystyle G=\langle \, \bigcup_{i=1}^{m} H_{i} \rangle$ and, for all $\displaystyle j$, $\displaystyle H_{j} \bigcap \, \langle \, \bigcup_{i \ne j} H_{i} \rangle = {\{1\}}$.