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Prove that every p-group is nilpotent.

How can I prove that groups with order a power of a prime are nilpotent?

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Kind of related to this question. Basically: You show that the group has non-trivial centre, then pass to the quotient G/Z(G). –  m_l May 4 '12 at 11:04
The post this was linked to did not actually contain a proof (the argument proposed by the OP was flawed); I've added a correct proof, using the lower central series, to my answer there. Note, however, that there are many ways of defining "nilpotent" (upper central series, lower central series, and a few others); you'll want to specify exactly what is your definition or what conditions you know are equivalent to nilpotency for this question to make sense. –  Arturo Magidin May 4 '12 at 18:35
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marked as duplicate by t.b., Brian M. Scott, anon, J. M., Martin Sleziak May 4 '12 at 11:22

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up vote 1 down vote accepted

Here is a link for the proof:

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