Transitive Group Action: Product of Stabilizers Not Equal to G?

[fireisland27]

Homework Statement

Let G be a group acting transitively on a set S. for a and b elements in S which are distinct, show that the product of the stabilizer of a and the stabilizer of b is not equal to G.

Homework Equations

The Attempt at a Solution

I was trying to use the orbit-stabilizer theorem and the fact that there is only one orbit of any element due to transitivity and somehow show that the product of the sizes of the stabilizers isn't equal to the size of G, but this doesn't seem to be going anywhere. I don't really know many theorems about group actions so I'm fairly lost as to how to find a solution. Can you point me in the right direction?

 

[arkajad]

If G acts transitively then there is a group element that transforms a into b. It's inverse transforms b into a. Can this group element be expressed as a product of elements that stabilize a or b? Try it. Remember that the stabilizer is a subgroup, therefor it contains with each element also its inverse.