# Fraction of tuples for iterated Lie bracket word in subring of finite Lie ring is at least as much as in whole ring

From Groupprops

## Statement

Suppose is a Lie ring word in letters that involves iterations of the Lie bracket only, and where every letter appears exactly once. For a finite Lie ring , consider the fraction:

Then, if is a subring of , we have:

The possibilities for include words such as (which define nilpotency class and words such as (which defines derived length ).

## Facts used

## Related facts

- Commuting fraction in subring of finite non-associative ring is at least as much as in whole ring
- Commuting fraction in subring of finite Lie ring is at least as much as in whole ring
- Associating fraction in subring of finite non-associative ring is at least as much as in whole ring
- Commuting fraction in subgroup is at least as much as in whole group