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

From Groupprops
Jump to: navigation, search

Statement

Suppose R is a finite Non-associative ring (?) (i.e., R is a not necessarily associative ring whose underlying set is finite). Suppose S is a subring of R. Then, the commuting fraction of S is at least as much as that of R.

In symbols, if CP(R) := \{ (x,y) \in R^2 \mid x * y = y * x \} and CP(S) = S^2 \cap CP(R), then:

\frac{|CP(S)|}{|S|^2} \ge \frac{|CP(R)|}{|R|^2}

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

Facts used

  1. Fraction of tuples satisfying groupy relation in subgroup is at least as much as in whole group

Related facts

Proof

The proof follows from fact (1), and the observation that the relation of commuting is groupy in both variables: in the sense that if we fix x, the set of y that commute with x form a subgroup of the additive group of R. (Note that since the relation is symmetric, groupiness in one variable is equivalent to groupiness in the other).