How many subgroups are there in Z/14Z?

Algebra by Geoffrey
  1. (5 points) How many a E Ss are there such that a(1) = 2, 0(2) = 1 and a is conjugate to (1 5)(2 3)?
  2. (5 points) G and G’ are groups. f : G —r G’ is a map. Define the set H = {(g, g’) E G x G’ig’ = f (g)}. Prove H is a subgroup of G x G’ if and only if f is a homomorphism.
  3. (5 points) How many subgroups are there in Z/14Z?
  4. (5 points) What is the order of the group Aut(Aut(Z/10Z))?
  5. (5 points) G is a group. If Igi = 2 for any non-identity g E G, prove that G is
  6. (5 points) R is the group of real numbers with addition. Q is the subgroup of rational numbers. r is a nonzero real number, denote rQ = {rq E Rig E Prove R/rQ = R/Q,

(5 points) f : G —r G’ is a homomorphism. K = ker(f). H is a subgroup of G, and H’ = {f (h) E       E H}, M ={gE f (g) EH’}. Prove that M = HK