Commuting fraction in subring of finite non-associative ring is at least as much as in whole ring
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 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.
- Commuting fraction in subring of Lie 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
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 , the set of that commute with form a subgroup of the additive group of . (Note that since the relation is symmetric, groupiness in one variable is equivalent to groupiness in the other).