Homework for Week 6

Part 1

  1. (Some of Rotman 2.52) True or false, with explanation.
    1. If H is a subgroup of K and K is a subgroup of G then H is a subgroup of G.
    2. G is a subgroup of itself.
    3. The empty set is a subgroup of G.
    4. The intersection of two cyclic subgroups of G is a cyclic subgroup.
    5. If X is an infinite set, then
      F = {w in SX such that w moves only finitely many elements of X}
      is a subgroup of SX.
    6. Every proper subgroup of S3 is cyclic. (A proper subgroup of a group G is a subgroup H other than G itself.)
    7. Every proper subgroup of S4 is cyclic.
  2. (Rotman 2.55) Give an example of two subgroups H and K whose union is not a subgroup.
  3. What is the order of σ ∙ τ if
    • σ = (12345) and τ = (678)
    • σ = (123456) and τ = (789)
    • σ = (123456) and τ = (7,8,9,10)
    • σ = (12345...k) and τ = (k+1, k+2, ..., k+n)
  4. Find all subgroups of Z/nZ for n=6, 10, 14, 15. What is the order of each? (Any conjectures?)
  5. Prove that every subgroup of a cyclic group is cyclic.
  6. (Rotman 2.40) Let G be a group and let g be an element of G with order m. If m=dt for some positive integer d, prove that gt has order d.

Part 2

  1. List all cosets of H in G, when
    1. H = <2>, G = integers mod 8
    2. H = <(123)>, G = S3
    3. H = <(12)(3)>, G = S3
  2. (Rotman 2.52vi) True or false with reasons: If H is a subgroup of G, then the intersection of two (left) cosets of H is a (left) coset of H.
  3. (Rotman 2.63)
    • Show that a left coset of <(12)(3)> in S3 may not be equal to a right coset of <(12)(3)> in S3; that is, find w in S3 so that w<(12)(3)> is not the same as <(12)(3)>w.
    • Let G be a finite group and let H be a subgroup of G. Prove that the number of left cosets of H in G is the same as the number of right cosets of H in G.
  4. If H and K are subgroups of a group G and if |H| and |K| are relatively prime, then prove that the intersection of H and K consists just of the identity element.
  5. What is the largest order of an element in Sn for n=1,2,3,...,10? What is an upper bound on the order of a subgroup of Sn for n=1,2,3,...,10?
  6. (Rotman 2.46) If G is a group with an even number of elements, prove that the number of elements in G of order 2 is odd. In particular, G must have some element of order 2.