Why does $N$ finite group, $M\unlhd N$ imply that $M$ contains a minimal normal subgroup of $N$?
If $M$ is itself minimal in $N$ we have finished. But if not, we know that there exists a minimal subgroup $P\stackrel{min}{\le}N$ s.t. $P\le M\le N$. But why, among such $P\;$'s, can we find one of them which is normal in $N$?
Thanks
While any subgroup of a finite group contains a minimal subgroup, a normal subgroup doesn't necessarily contain a subgroup that is both minimal and normal. Such a subgroup would be called a normal minimal subgroup. If you reverse the two adjectives and speak instead of minimal normal subgroups you are talking about something else: you are talking about subgroups which are minimal among normal subgroups. Now the fact should be clear: the collection of nontrivial normal subgroups of a finite group will be finite, nonempty and partially ordered (by inclusion) + in any finite partially ordered set, every element is greater than some minimal element.