Lower central series members are divisibility-closed in nilpotent group

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Statement

Suppose G is a nilpotent group. Then, all members of the lower central series of G are divisibility-closed subgroups of G.

Related facts

Similar facts

Opposite facts

Applications

Facts used

  1. Equivalence of definitions of nilpotent group that is divisible for a set of primes: We are interested in the (1) implies (4) part of the equivalence for the "nilpotent group and a prime" case: if G is a p-divisible nilpotent group, then each of the quotients \gamma_i(G)/\gamma_j(G) is also p-divisible for positive integers i < j.

Proof

The proof follows from the (1) implies (4) implication of Fact (1). Specifically, if we want to show that \gamma_m(G) is divisibility-closed in G, we make two cases:

  • G has class less than m: In this case, \gamma_m(G) is trivial, and there is nothing to prove.
  • G has class m or higher: In this case, if c is the class of G, set i = m, j = c + 1.