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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.