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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.

  1. G1 = (a | a^17=e).

  2. G2 = (a,b | a3=e, ab-1=a-1).

  3. 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.

2014-06-17 11:57