Homework for Week 5

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 w^r is the identity permutation.
   2. Prove that r is the least positive integer k such that w^k is the identity permutation. (This means that r is the order of w.)
   3. If r is prime, prove that every power w^k is either the identity permutation or a p-cycle (namely, when w^k 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 (g^j)(g^k)=g^j+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.)

Due Friday 2/24.