Prove that is an infinite group as follows. Let be a prime congruent to 1 mod 3 and let denote the nonabelian group of order . Let with and . Prove that and have order 3. Deduce that is a homomorphic image of , and from this deduce that is infinite, using the fact that there are infinitely many primes congruent to 1 mod 3. [Note that every nonidentity element of has order 3 or .]
Let be a prime congruent to 1 mod 3, and let , where , , and for some order 3 automorphism .
In particular, for some mod , and mod . Note that mod .
Now we compute : . Thus .
Similarly, it is easy to see that .
Now let and be defined by , . Clearly and generate . By the universal property of free groups there exists a unique group homomorphism extending ; moreover, because (as computed above) and satisfy , there exists a unique group homomorphism such that . Since is surjective, so is . Thus is a homomorphic image of .
Since there are infinitely many primes congruent to 1 mod 3, cannot have finite order.