Question: I would like to know how to find the "smallest subgroup of G"  (Thrm. 5.14, p.52)  using an example.

Answer: You wrote "Thrm. 5.14, p.52" but I think you mean Thm. 5.17, p. 53.

What "smallest" means is this: If K is a subgroup of G and a is in K, then H is a subset of K.  You might try proving this---it would be a good test question!

But you asked for an example.  So, for argument's sake, let's let G=S_4.

Define f by f(1)=2, f(2)=3, f(3)=4, and f(4)=1.  Then f is in S_4.  We would write f=

1 2 3 4

2 3 4 1

Let's let H be the smallest subgroup of S_4 that contains f.  Let's see if we can figure out what H is.

Well, we know H contains f.

Also, H is a subgroup, so H contains e, the identity.

Also, H is a subgroup, so H is closed under the group law.  So f^2 is in H.

So far, we have three elements in H, namely f, e, and f^2.  Can we find any others?

Well, H is also closed under inverses, so f^(-1) is also in H.

If you think about it, you'll see that H must contain f^n for all integers n.

This is exactly the definition of the subgroup generated by f.

In our example, you will find that H={e, f, f^2, f^3}.