# Commuting fraction in subring of finite Lie ring is at least as much as in whole ring

From Groupprops

## Contents

## Statement

Suppose is a finite Lie ring (?) (i.e., is a Lie ring whose underlying set is finite). Suppose is a subring of . Then, the commuting fraction of is at least as much as that of .

In symbols, if and , then:

In fact, the result also holds if is simply an *additive* subgroup of and not a subring.

## Facts used

## Related facts

- Fraction of tuples for iterated Lie bracket word in subring of finite Lie ring is at least as much as in whole ring
- Commuting fraction in subring of finite non-associative ring is at least as much as in whole ring
- Associating fraction in subring of 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

## Proof

The proof uses fact (1), and the observation that the relation of *commuting* in a Lie ring is a groupy relation.