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.
G1 = (a | a^17=e).
G2 = (a,b | a3=e, ab-1=a-1).
G3 = (a,b,c | a2=b2=c2=e, ab=c, bc=a, ca=b).
2. A homomorphic problem
Let G be a group with |G| = 2n 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.
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 (n2+1). Hint: consider n2 mod p and apply Fermat's Little Theorem.