PineWiki
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.
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 (n2+1). Hint: consider n2 mod p and apply Fermat's Little Theorem.
