Penultimate term of lower central series is abelian in nilpotent group of class at least three

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Statement

If G is a Nilpotent group (?) and the Nilpotence class (?) of G is an integer m \ge 3, then the (m-1)^{th} term of the Lower central series (?) of G is Abelian. In particular, it is an Abelian characteristic subgroup of G.

Related facts

Corollaries

Breakdown of corresponding statement for upper central series

The second term of the upper central series of a finite nilpotent group need not be abelian. Further information: Second term of upper central series not is abelian, Upper central series may be tight with respect to nilpotence class

Facts used

  1. Second half of lower central series of nilpotent group comprises Abelian groups: If G is a group of nilpotence class m, and k \ge (m + 1)/2, the member G_k of the lower central series of G is Abelian.

Proof

The proof follows directly from fact (1), and the observation that for m \ge 3, m - 1 \ge (m + 1)/2.

References

Textbook references