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$p$-groups are locally nilpotent groups? How can we show it? I know that a finite $p$-group is nilpotent. If in general the answer is "no", then what about a FC $p$-group?

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    Thanks. Do You know something about $FC\space p$-groups? Are they locally nilpotent groups?2012-10-12

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In periodic $FC$-groups Dietzmann's lemma forces $G$ to be locally nipotent.