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

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.

Note that this is the same list as for nilpotent groups, but the (4) implies (1) implication is missing.

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

The duality is as follows:

Corollaries

 * Derived subgroup is divisibility-closed in nilpotent group
 * Lower central series members are divisibility-closed in nilpotent group

Failed generalizations

 * Residually nilpotent group with abelianization that is divisible by a prime need not be divisible by that prime
 * Solvable group with abelianization that is divisible by a prime need not be divisible by that prime

Facts used

 * 1) uses::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) uses::Divisibility is central extension-closed

Proof
We first 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.