Lecture 12, MTH 204A (Section B)

The action of a group G on itself gives rises to the following natural actions:
1. action of K on G/H, where H and K are subgroups of G, and
2. action of G on the power set P(G).

As special instances of (1), using the Orbit-Stabilizer theorem, we prove the following in this lecture:
1. any subgroup H with index as the smallest prime dividing |G| is normal
2. Sylow's 2nd theorem.

Next, we analyse the action of G on P(G). We prove that the order of the stabiliser of any subset A is a divisor of order of A. Using this, we prove the first and third theorem of Sylow in this lecture.