Stabilizer

If we have a group G acting on a set Ω, the stabilizer of an element ω in Ω is defined by

Stab(ω) := {g in G | ωg = ω}

i.e. the set of elements of G which "leave ω alone". Note that a stabilizer is always a subgroup of G. The concept of a stabilizer is closely related to that of an orbit - see in particular the orbit-stabilizer theorem.

Simple example of a stabililizer - let G := S_n - the symmetric group of permutations of {1, 2, ..., n} - and let Ω := {1, 2, ..., n}.

Then Stab(n) = {g in G which don't move n} = {permutations of {0, 1, ..., n-1} } = S_n-1