Math 817: Problem set 10
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 12, 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.)
- Let n be an integer, at least 3.
- (a) Determine the center Z(Dn) of Dn.
[Hint: Consider n even and n odd.]
- (b) Let G and H be groups. Prove that Z(GxH) = Z(G)xZ(H).
- (c) Prove that D2n is isomorphic to
DnxZ(D2n) if and only if n is odd.
- Given a 2x2 integer matrix P,
define a map TP: ZxZ -> ZxZ
by TP(v) = Pv, where we regard elements of
ZxZ as column vectors with integer entries.
- (a) Show that TP is a homomorphism of groups.
- (b) Show that TP is injective if and only if det(TP) is not 0.
- (c) Show that TP is surjective if and only if |det(TP)| = 1.
- (d) Let h: ZxZ -> ZxZ be an isomorphism
of groups. Show that h = TP for some 2x2 integer matrix P of determinant 1 or -1.
- Let G be a group and consider the set S = G. Define a map a: GxS -> S
by a((g,s)) = gsg-1, using the group multiplication and inversion.
- (a) Show that this defines an action of G on S = G (i.e., on itself).
- (b) Let h: G -> Perm(S) be the homomorphism associated to this action.
Show that h(G) is a subgroup of Aut(G).
- (c) Show that ker h = Z(G).
- Do Problem #10, on p. 190. For parts (a) and (b), it is enough
to give just the answers; no justifications are needed. For part (a),
your answer should be the complete list of all 2x2 matrices
in the point group of G. For part (c), give a brief explanation
of how you get your answer.
- Do Problem #12 on p. 193.
- Let G be a group acting on a set S. We say the action is trivial
if gs = s for all s in S and all g in G. Now let H be a subgroup of G.
Let H act on the left cosets of G/H by left multiplication; i.e.,
if h is in H and xH is in G/H, then h(xH) = (hx)H. Show that the action
is trivial if and only if H is a normal subgroup of G.