Math 817: Problem set 12
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, December 3, 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 H be a subgroup of prime index p in a finite
group G. Use the generalized
Cayley's theorem to prove that H is normal, if
p is the smallest prime dividing the order of G.
 Let G be the automorphism group of D_{5}.
 (a) Determine G.
 (b) Show that G is not abelian.
 (c) Find an element f of G of order 4.
 (d) In the terminology of Problem 2 from Homework 2,
show that f^{2} = a(g) for some g in D_{5}.

 (a) Find a representative of each conjugacy class
of A_{5}, and determine the order of the class.
(No justification is needed.)
 (b) Determine the orders of possible normal subgroups of
A_{5}.
 (c) Use your answer to (b) to determine whether
A_{5} is a simple group.
 Do Problem 8 on p. 196.
 Let G be a finite group whose order is divisible
by a prime p. The point of this problem is to prove
Cauchy's theorem, that G must have an element of order p.
So let the cyclic group C_{p} =
{1, c, c^{2}, ..., c^{p1}} act
on the direct product GxGx...xG = G^{p} (of p copies of G)
by cyclic permutations; i.e., if (g_{1}, ..., g_{p})
is an element of G^{p}, then, for example,
c(g_{1}, g_{2}, ..., g_{p}) =
(g_{p}, g_{1}, ..., g_{p1}).
Let U = {(g_{1}, g_{2}, ..., g_{p}) in G :
g_{1}g_{2}...g_{p} = 1_{G}} 
{(1_{G}, ..., 1_{G})}; i.e., U is the subset of
G^{p} of all sequences of p elements of G (other than
the constant sequence whose terms are all the identity)
such that the product
of the terms is the identity.
 (a) Show that U is a union of orbits under the action of
C_{p} on G^{p}, hence that
C_{p} acts on U.
 (b) Show that p doesn't divide U.
 (c) Use the Fixed Point Theorem (p. 199) to
show that G has an element of order p.
[Hint: Show that an element
(g_{1}, g_{2}, ..., g_{p})
of U is fixed under
the action of C_{p} if and only if
all of the entries g_{i} are equal.]