Prove that in a nilpotent group every normal subgroup of prime order is contained in the center.
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2026-03-25 19:06:57.1774465617
Prove that in a nilpotent group every normal subgroup of prime order is contained in the center.
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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 $\;1\neq N\lhd G\implies N\cap Z(G)\neq 1\;$