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

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Suppose R is a finite Lie ring (?) (i.e., R is a Lie 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] = 0 \} 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


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