# 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

## Contents

## Statement

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

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

- is -divisible.
- The abelianization of is -divisible.
- For every positive integer , the quotient group of successive members of the lower central series is -divisible.
- For any two positive integers , if denote respectively the and members of the lower central series of , then the quotient group is -divisible.

### For a nilpotent group and a prime

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

- is -divisible.
- The abelianization of is -divisible.
- For every positive integer , the quotient group of successive members of the lower central series is -divisible.
- For any two positive integers , if denote respectively the and members of the lower central series of , then the quotient group is -divisible.

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

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

- is -divisible.
- The abelianization of is -divisible.
- For every positive integer , the quotient group of successive members of the lower central series is -divisible.
- For any two positive integers , if denote respectively the and members of the lower central series of , then the quotient group is -divisible.

### For a nilpotent group and a set of primes

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

- is -divisible.
- The abelianization of is -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

- 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

- Divisibility is quotient-closed
- Each successive quotient of the lower central series is a homomorphic image of a tensor power of the abelianization via the iterated commutator map.
- 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 by its derived subgroup.

### (2) implies (3)

**Given**: A group such that the abelianization of is -divisible. A positive integer .

**To prove**: is -divisible.

**Proof**: This follows directly from the given and Fact (2): is the homomorphic image of a tensor power of the abelianization of via the iterated commutator map.

### (3) implies (4)

**Given**: A group such that each of the successive quotients of lower central series members is -divisible. Positive integers .

**To prove**: The quotient is -divisible.

**Proof**: We induct on , 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 and (where has nilpotency class ) to get the result.