# 1. Generators and relations

Below are some presentations of groups given by generators and relations. For each group, compute the number of elements, and prove that your count is correct.

G

_{1}= (a | a^17=e).G

_{2}= (a,b | a^{3}=e, ab^{-1}=a^{-1}).G

_{3}= (a,b,c | a^{2}=b^{2}=c^{2}=e, ab=c, bc=a, ca=b).

# 2. A homomorphic problem

Let G be a group with |G| = 2^{n} and let f be a surjective homomorphism from G to H. Prove that if |H| > 1, then |H| is even. *Note new assumption that |H| > 1 added 2004-11-30.*

# 3. Triskaidekainversia

Compute x^{-1} for each x in Z^{*}_{13}.

# 4. A little problem

Show that if p is prime and (p-1)/2 is odd, there is no number n such that p divides (n^{2}+1). Hint: consider n^{2} mod p and apply Fermat's Little Theorem.