## Math 417 Homework 9: Due Friday April 4

*Instructions*: You can discuss these problems with others,
but write up your solutions on your own (i.e., don't just
copy someone else's solutions, else the feedback I give
you won't help you much). Please be neat and write in full sentences.

Do four of the five problems. Justify your answers.

- [1] Problem #14 on p. 145.

- [2] Problem #16 on p. 145.

- [3] Problem #30 on p. 145.

- [4] Problem #38 on p. 145.

- [5] Let G be a subgroup of S
_{n}, thus
G is a group of permutations of the set A = {1, 2, ..., n},
and define the function f : G -> **N** by
f(g) = n_{g}, where n_{g} is the
number of elements i of {1, 2, ..., n} such that
g(i) = i.
- (a) Prove that any two orbits of G in A are either equal
or disjoint.
- (b) If g
_{1}, ... , g_{r} are the distinct elements
of G, prove that f(g_{1}) + ... + f(g_{r}) = r|G|,
where r is the number of orbits of G in A. [Hint: First try to
do the case that r = 1.]