# Math 817: Problem set 2

*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 10, 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 a = 5723 and let b = 5959.
- (a) Find a generator for the subgroup a
**Z** + b**Z**
of **Z**.
- (b) What is the least positive integer c such that
there is a solution (x, y) in integers for the equation ax + by = c?
Justify your answer, and find a solution (x, y) for that c.

- For this problem, we start with some terminology.
Let G be a group. Let g be an element of g; then we have
a mapping f
_{g}: G -> G called *conjugation*
by g, defined by f_{g}(x) = gxg^{-1}, for all x in G.
An element of the form gxg^{-1} is called
the *conjugate* of x by g.
The *center* of G, denoted Z(G), is the subset
{x in G : xy = yx for all y in G} of G.
- (a) Let Aut(G) denote the set of all automorphisms of G.
Show that Aut(G) is a subgroup of the group Perms(G) of all
permutations of G (hence Aut(G) is a group).
- (b) The book shows that f
_{g} is an automorphism.
Show that a: G -> Aut(G) defined by
a(g) = f_{g} is a homomorphism with kernel Z(G)
(hence Z(G) is a normal subgroup).
- (c) If H is a subgroup of G, then f
_{g}(H)
is traditionally denoted gHg^{-1}, called
the *conjugate* of H by g. Since
gHg^{-1} is the image of
H under the homomorphism f_{g}, we know
that gHg^{-1} is a subgroup of G.
Show that H is normal if and only if
gHg^{-1} = H for all g in G.

- Do problem 7 on p. 71: in the notation on p. 11,
let A be the 2x2 matrix I
_{2} + e_{1,2}
and let B = I_{2} + e_{2,1}.
Show that A is conjugate to B (or, equivalently, vice versa)
in GL_{2}(**R**),
but they are not conjugate in SL_{2}(**R**).
- Let G = (g) be a finite cyclic group of order n.
- (a) Let i be a positive integer. Show that
(g
^{i}) = (g^{d}), where d = gcd(i, n),
and that |g^{i}| = n/d.
- (b) Show that there is a one to one correspondence between
positive integer divisors of n and subgroups of G.
[Hint: Justify the fact that every subgroup of G is cyclic and apply (a).]
- (c) Show that a finite group H is cyclic if and only if
there is a unique cyclic subgroup
of H of order j for each divisor j > 0 of |H|.

- Let f: X -> Y be a mapping of sets. Let A be a subset of X
and let B be a subset of Y.
- Show that f(f
^{-1}(B)) is a subset of B, and that
f(f^{-1}(B)) = B if f is surjective.
- Formulate and prove an analogue of (a) involving injectivity.