Instructions: Write up any four of the following problems (but
you should try all of them, and
in the end you should be sure you know how to do all of them).
Your write ups are due Friday, September 17, 2004.
Each problem is worth 10 points, 9 points for correctness
and 1 point for communication. (Your goal is not only
to give correct answers but to communicate
your ideas well. Make sure you use good English,
so proofread your solutions.
Once you finish a solution, you
should restructure awkward sentences, and strike out
anything that is not needed in your approach to the problem.)
(a: 2 points) Translate Hilbert's quote on p. 78:
Immer mit den einfachsten Beispielen anfangen.
(b: 8 points) (This problem is a modification of #9, p. 73.)
Let R be the smallest equivalence relation on the reals R
such that R contains {(x,y) : x - y = 1}. Sketch R.
Show that the equivalence classes are cosets with respect to
a certain subgroup, determine that subgroup, and show that it is the kernel of
a homomorphism of R to some group G.
Let H and K be subgroups of a group G.
If H and K are normal,
show that both HK and the intersection I of H with K
are subgroups which are normal in G.
Let G be a group.
If x^{2} = e_{G} for all elements x in G, show that
G is abelian.
If |G| = 4, show that G is abelian.
(a: This is #4, p. 73. 4 points) Let R and R' be equivalence relations
on a set S. Must their intersection I be an equivalence relation?
Give a proof or a counterexample. Must their union U?
Again, give a proof or a counterexample.
(b: This is #8, p. 73. 6 points) Fill in a table in which you indicate
for each of the following relations on the reals R
which of the properties reflexivity, symmetry and transitivity
hold for the given relation. (You do not need to justify your answers.)
R
reflexive?
symmetric?
transitive?
(i) R = {(s, s) : s is in R}
(ii) R = { }
(iii) R = {(x, y) : y = 0}
(iv) R = {(x, y) : xy + 1 = 0}
(v) R = {(x, y) : x^{2}y - xy^{2} - x + y = 0}
(vi) R = {(x, y) : x^{2} - xy + 2x - 2y = 0}
This is #10, p. 74.
(a) Show that every subgroup of index 2 is normal.
(b) Give an example of a subgroup of index 3
which is not normal.