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Groupprops β

Nilpotent group that is divisible for a set of primes

Definition

Suppose G is a group and \pi is a set of primes. We say that G is a \pi-divisible nilpotent group if it satisfies the following equivalent conditions:

  1. G is nilpotent and \pi-divisible.
  2. G is nilpotent and the abelianization of G is \pi-divisible.
  3. G is nilpotent and for every positive integer i, the quotient group \gamma_i(G)/\gamma_{i+1}(G) of successive members of the lower central series is \pi-divisible.
  4. G is nilpotent and for any two positive integers i < j, if \gamma_i(G),\gamma_j(G) denote respectively the i^{th} and j^{th} members of the lower central series of G, then the quotient group \gamma_i(G)/\gamma_j(G) is \pi-divisible.

Equivalence of definitions

Further information: Equivalence of definitions of nilpotent group that is divisible for a set of primes