Lecture 16, MTH 204A (Section B)

We begin with a theorem that provides us with a set of conditions that is sufficient to guaranty when two semidirect products of two given groups H and K will be isomorphic. Using this, we complete the classification of groups of order pq, for p and q primes and p is less than q. Next we talk about the following applications of Sylow's theorems: 1. In every group of order 30, Sylow 3 and 5 subgroups are normal. 2. If G is a group of order 12 then one of the Sylow 2 and 3 subgroups will be normal. Furthermore, if there are more than one Sylow 3 subgroups then G is isomorphic to A_4. 3. If p and q are distinct primes then any group of order p^2q, at least one Sylow subgroup will be normal. In fact, except the case p=2 and q=3, the Sylow subgroup corresponding to the bigger prime will always be normal. 4. If a group of order 60 has more than one Sylow 5 subgroup then it has to be simple. The simplicity of A_5 also follows from that.