# Associating fraction in subring of finite non-associative ring is at least as much as in whole ring

From Groupprops

## Contents

## Statement

Suppose is a finite Non-associative ring (?) (i.e., is a not necessarily associative ring whose underlying set is finite). Suppose is a subring of . Then, the associating 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

- Associativity is a groupy relation on the additive group of a non-associative ring
- Fraction of tuples satisfying groupy relation in subgroup is at least as much as in whole group

## Related facts

- Commuting fraction 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
- Commuting fraction in subgroup is at least as much as in whole group

## Proof

The proof follows from facts (1) and (2).