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

From Groupprops
Jump to: navigation, search
This article gives a proof/explanation of the equivalence of multiple definitions for the term nilpotent group that is divisible for a set of primes
View a complete list of pages giving proofs of equivalence of definitions

Statement

For an arbitrary (not necessarily nilpotent) group and a prime

For a group G and a prime number p, we have the implications (1) implies (2) implies (3) implies (4):

  1. G is p-divisible.
  2. The abelianization of G is p-divisible.
  3. For every positive integer i, the quotient group \gamma_i(G)/\gamma_{i+1}(G) of successive members of the lower central series is p-divisible.
  4. 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 p-divisible.

For a nilpotent group and a prime

The following are equivalent for a nilpotent group G and a prime number p:

  1. G is p-divisible.
  2. The abelianization of G is p-divisible.
  3. For every positive integer i, the quotient group \gamma_i(G)/\gamma_{i+1}(G) of successive members of the lower central series is p-divisible.
  4. 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 p-divisible.

For an arbitrary (not necessarily nilpotent) group and a set of primes

For a group G and a set of prime numbers \pi, we have the implications (1) implies (2) implies (3) implies (4):

  1. G is \pi-divisible.
  2. The abelianization of G is \pi-divisible.
  3. 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. 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.

For a nilpotent group and a set of primes

The following are equivalent for a nilpotent group G and a set of prime numbers \pi:

  1. G is \pi-divisible.
  2. The abelianization of G is \pi-divisible.
  3. 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. 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.

Related facts

Dual fact

For more on the background, see subgroup-quotient duality for groups.

The dual fact is equivalence of definitions of nilpotent group that is torsion-free for a set of primes

The duality is as follows:

Concept of the torsion-free side Concept on the divisible side
torsion-free divisible
center abelianization
inner automorphism group derived subgroup
upper central series lower central series

Corollaries

Failed generalizations

Facts used

  1. Divisibility is quotient-closed
  2. Each successive quotient of the lower central series is a homomorphic image of a tensor power of the abelianization via the iterated commutator map.
  3. Divisibility is central extension-closed

Proof

We show the implications for an arbitrary group and a single prime. We then show the (4) implies (1) implication that requires the whole group to be nilpotent. The results for sets of primes follow immediately from the results for a prime, so their proofs are not given separately.

(1) implies (2)

This follows directly from Fact (1), and the observation that the abelianization is the quotient of G by its derived subgroup.

(2) implies (3)

Given: A group G such that the abelianization of G is p-divisible. A positive integer i.

To prove: \gamma_i(G)/\gamma_{i+1}(G) is p-divisible.

Proof: This follows directly from the given and Fact (2): \gamma_i(G)/\gamma_{i+1}(G) is the homomorphic image of a tensor power of the abelianization of G via the iterated commutator map.

(3) implies (4)

Given: A group G such that each of the successive quotients of lower central series members is p-divisible. Positive integers i < j.

To prove: The quotient \gamma_i(G)/\gamma_j(G) is p-divisible.

Proof: We induct on j - i, using the given data for the base case and inductive step, and using Fact (3) in order to execute the induction step.

(4) implies (1): only for nilpotent groups

For this, set i = 1 and j = c + 1 (where G has nilpotency class c) to get the result.