# Existence of abelian ideals of small prime power order in nilpotent Lie ring

From Groupprops

## Statement

Suppose is a prime number and is a nilpotent Lie ring of order . Then, if is a nonnegative integer such that (i.e., ), has an Abelian ideal (?) of order .

## Related facts

### Similar facts

- Existence of abelian normal subgroups of small prime power order (see more facts related to that)

## Facts used

- Lower bound on order of maximal among abelian ideals in terms of order of nilpotent Lie ring
- Finite nilpotent Lie ring implies every ideal contains ideals of all orders dividing its order

## Proof

*Outline*: We use Fact (1) to show that there is an abelian ideal of order at least , and then use Fact (2) to find within that an abelian ideal of order exactly .