Orbit-Stabilizer Theorem & Applications

A short expository paper for Stanford's Writing-in-the-Major program (Math 120). The Orbit-Stabilizer theorem connects the structure of a transitive group action to the coset structure of a stabilizer subgroup — an algebraic identity that's almost trivial to state and surprisingly powerful in application.

The paper proves the theorem (constructing the equivariant bijection between the orbit and the coset space), applies it to the conjugation action of a group on itself to deduce the sizes of conjugacy classes and centralizers, and uses these tools to prove that every finite p-group has a nontrivial center.

For a portfolio context, it's a window into the abstract-algebra side of the math major — small but written carefully.