In algebra, the bicommutant of a subset S of a group is the commutant of the commutant of that subset. It is also known as the double commutant or second commutant and is written S′ ′.
The bicommutant of S always contains S. So S′ ′ ′ = (S′ ′) ′ ⊆ (S′). On the other hand, S′ ⊆ (S′) ′ ′ = S′ ′ ′. So S′ = S′ ′ ′, i.e. the commutant of the bicommutant of S is equal to the commutant of S. By induction, we have:
- S′ = S′ ′ ′ = S′ ′ ′ ′ ′ = ... = S(2n-1) = …
and
- S ⊆ S′ ′ = S′ ′ ′ ′ = S′ ′ ′ ′ ′ ′ = ... = S(2n) = …
for n > 1.