Math 817: Problem set 11
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, November 19, 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.)
- Each part of this problem might be useful for the next!
- (a) Let Z be the center of a group G. If G/Z is cyclic,
show that G is abelian (and hence G=Z).
- (b) If G is a nonabelian group of order 10,
show that the class equation for G must be 10 = 1 + 2 + 2 + 5.
- (c) If G is a nonabelian group of order 10, show that G is
isomorphic to D5. [Hint: conclude from (b)
that G has an element x of order 5 and an element y of order 2.
Show that yxy = x4, and show that this uniquely
determines the multiplication table of G. Hence, up to
isomorphism, there is a unique nonabelian group of order 10.]
- Prove Proposition 1.12 on p. 199.
- Let G be the group of order preserving symmetries of
the set X = {(a,0,1), (a,0,-1), (-a,1,0), (-a,-1,0)}, where
a = 2-1/2. Note that X is the set of vertices of
a regular tetrahedron T centered at the origin. Thus G is
the tetrahedral subgroup of SO3. Determine the
set S of poles of G (describe them explicitly, either in terms
of T or by giving their coordinates), determine |S|, and
explicitly determine the orbits of G acting on S
(i.e., which poles are in which orbits). You do not
need to justify your answers (except to yourself!).
- (Burnside's formula)
Let G = {g1, ..., gr} be a group
of order r acting on a finite set S. Let n be the number of orbits.
For each g in G, let Sg = {s in S : gs = s}.
Prove that n|G| = |Sg1| + ... + |Sgr|.
[Hint: Look at C = {(g,s) : g is in G, s is in S, gs = s}.]
- (A generalization of Cayley's theorem)
Let H be a subgroup of a group G of finite index [G:H] = n.
Let G act on the left cosets G/H of H in G by left multiplication.
Show there exists a homomorphism f: G -> Sn such that
ker f is contained in H and
every normal subgroup of G which is contained in H is contained
in ker f (i.e., ker f is the largest normal subgroup of
G contained in H).
- Let L = {me1 + ne2: m and n are integers}.
Note that L is an additive subgroup of R2.
Let v = (0.1, 0.01)t and let G = Sym(v + L) be the
rigid motions of the plane which preserve the coset v + L.
- Determine the point group of G (actually list the matrices
explicitly; no proof is needed).
- Determine the translation subgroup LG of G
(no proof is needed, but be clear as to which vectors are in
LG).
- Artin shows that the point group acts on LG (see p. 169),
but goes on to say that ``it is important to note that ... G
need not operate on LG.'' Verify this by exhibiting
an element g of G and a vector w in LG, such that
gw is not in LG.