We’re all familiar with Cayley’s theorem: every group is isomorphic to a subgroup of In the finite case, every finite group of order is isomorphic to a subgroup of . Now let’s see some applications of Cayley’s idea.

**Problem 1**. Let be a group and let be a subgroup of with Prove that there exists a normal subgroup of such that and

**Solution**. Let be the set of all left cosets of in . Define

by Then see that Thus is well-defined. It is easy to show that is a group homomorphism and So is isomorphic to a subgroup of Thus

**Problem 2**. Let be a finite group and let be the smallest prime divisor of Let be a subgroup of with Prove that is a normal subgroup of

**Solution**. By Problem 1, there exists some normal subgroup such that Therefore for some integer . Thus But because no prime divisor of is smaller than So and hence because

A trivial result of Problem 2 is that if is a subgroup of a group and then is normal in Also if is a prime number and is a group of order then every subgroup of of order is normal.

**Problem 3**. Prove that if is an infinite simple group and is a proper subgroup of then

**Solution**. Suppose to the contrary that . Then, by Problem 1, contains some normal subgroup with In particular But is simple and , because Hence and so which is a contradiction!

**Problem 4**. Let and let be a proper subgroup of Prove that

**Solution**. Let By Problem 1, has a normal subgroup which is contained in But is simple because Thus and hence, by Problem 1, which is not possible unless

**Example**. By Problem 4, has no subgroup of orders