I have encountered a few problems regarding the minimal subgroups of a finite group $G$. Any references and/or answers regarding the following questions will be very welcome.
1)If $G$ is a finite group, what can be said about its minimal normal subgroups (under inclusion)?
2) Is there a criterion to decide when an element $g \in G$ belongs to such a minimal subgroup?
3) Is the product of all minimal normal subgroups of a given finite group $G$ the entire group $G$?
Best Answer
Try reading the planetmath.org article on the socle of a group.