Homework for Week 5

Part 1

  1. (Rotman 2.25, 2.29iii) Consider the permutation w = (1 2 3 ... r)(r+1)(r+2)(r+3)...(n) in cycle notation.
    1. Prove that wr is the identity permutation.
    2. Prove that r is the least positive integer k such that wk is the identity permutation. (This means that r is the order of w.)
    3. If r is prime, prove that every power wk is either the identity permutation or a p-cycle (namely, when wk is written in cycle notation, it has one cycle with exactly p elements, and the rest are cycles with just one element each).
  2. What are the orders of each unit mod n, for n=2,3,4,5,6,7,8? (You may use an earlier homework assignment.)
  3. (Rotman, parts of 2.36) True or false, with explanation.
    1. The binary operation on integers defined by m*n = mn is an associative operation.
    2. Every group is abelian.
    3. The set of all positive real numbers is a group under multiplication.
    4. The set of all positive real numbers is a group under addition.
    5. Let G be a group. For all elements a, b in G, the element a*b*a-1*b-1 is the identity e.
    6. Let G be a group. For all elements a, b in G, the element (ab)n = an bn for all nonzero integers n.
    7. Complex conjugation permutes the roots of every polynomial having real coefficients.
  4. Left-inverses equal right-inverses: Prove that if g,h are elements of a group G and gh=e then hg=e.
  5. Cancellation: Prove that if g,h,h' are elements of a group G and gh = gh' then h=h'.
  6. If g is an element of a group G then prove that (gj)(gk)=gj+k for all integers j,k (including negative integers!).
  7. If g,h are elements of a group G and gh=hg then prove that (gh)i = gihi for all nonnegative integers i.
(When in doubt about what you can assume, prove more rather than less.)

Part 2

  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.