Second half of lower central series of nilpotent group comprises abelian groups

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Statement

Suppose G is a nilpotent group of nilpotency class c. Define the lower central series of G as follows:

\gamma_1(G) = G, \qquad \gamma_{m+1}(G) = [\gamma_m(G),G].

Then, for k \ge (c + 1)/2, \gamma_k(G) is an abelian group, In particular, \gamma_k(G) is an abelian characteristic subgroup.

Facts used

  1. Lower central series is strongly central: This states that [\gamma_m(G),\gamma_n(G)] \le \gamma_{m+n}(G).

Applications

Breakdown for upper central series

The first half of the upper central series of a nilpotent group need not comprise Abelian groups. In fact, even the second term of the series need not be Abelian, however large the nilpotence class. More specifically:

Proof

Given: A group G of nilpotency class c.

To prove: \gamma_k(G) is Abelian for k \ge (c + 1)/2.

Proof: By fact (1), [\gamma_k(G), \gamma_k(G)] \le \gamma_{2k}(G) \le \gamma_{c+1}(G), and since G has class c, \gamma_{c+1}(G) is trivial. Thus, [\gamma_k(G), \gamma_k(G)] is trivial, and thus, \gamma_k(G) is an abelian group.