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

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 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

## 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.