Prove that in a nilpotent group every normal subgroup of prime order is contained in the center.
Solution 1:
Hints:
Using for example the upper central series
$$1=Z_0\lhd Z_1:=Z(G)\lhd Z_2\lhd\ldots\lhd Z_n=G$$
show that $\;1\neq N\lhd G\implies N\cap Z(G)\neq 1\;$