

Math 6120 Modern Algebra 

Homework: 

Assignment 1:
Assigned 9/14/2004
Section 1.2 #17
Section 1.6 #14, 17

Section 1.2 #17. Let
X_{2n}= < x,y  x^n = y^2 = 1, xy=yx^2 >
(a) Show that if n=3k then X_{2n} has order 6, and it has the same
generators and relations as D_6 when x is replaced by r and y by s.
(b) Show that if (3,n) = 1 then x satisfies the additional relation :
x=1. In this case deduce that X_{2n} has order 2. [Use facts that
x^n
=1 and x^3 = 1]
Section 1.6 #14 Let G and H be groups and let \hi : G > H be a
homomorphism. Define the kernel of \phi to be { g \in G  \phi(g)
=
1_H } (so the kernel is the set of elements in G shich map to the
identity of H, ie.e is the fiber over the identity of H). Prove
that
the kernel of \phi is a subgroup of G. Prove that \phi is injective if
and only if the kernel of \phi is the identity subgroup of G.
#17 Let G be any group. Prove that the map from G to itself
defined
by g > g^{1} is a homomorphism if and only if G is abelian.

Assigned 9/21/2004
Section 1.7 #10 (also understand 8,9), 18, 19, 21
Section 2.1 #10

Section 1.7 #10
(#8 The symmetric group acts on the set of all subsets of cardinality k
by the action \sigma { a_1, ..., a_k } = { \sigma(a_1), ...,
\sigma(a_k) }. Prove it is a group action. #9 The symmetric
group
also acts on the set of ktuples of the numbers 1 through n...it is not
clear if they are allowing the entries in the ktuples to repeat in
this question so take a guess and make your answer clear)
With reference to the previous two exercises determine
(a) for which values of k the action of S_n on kelement subsets is
faithful
(b) for which values of k the action of S_n on ktuples is faithful
#18 Let H be a group acting on a set A. Prove that the relation
\sim on A defined by
a \sim b if and only if a = hb for some h \in H
is an equivalence relation. (for each x in A the equivalence
class of
x under \sim is called the orbit of x under the action of H. The
orbits under the action of H partititon the set A).
#19 Let H be a subgroup of of the finite grou G and lt H act on G
(here A=G) by left multiplication. Let x \in G and let O_x be the
orbit of x under the action of H. Prove that the map
H > O_x defined by x > hx is a bijection (hence all
orbits
have cardinality H). From this and the preceeding exercise
deduce
Lagrange's Theorem:
if G is a finite group and H is a subgrou of G then H divides G.
#21 Show that the group of rigid motions of a cube is isomorphic to S_4.
Section 2.1 #10 (a) Prove that if H and K are subgroups of G then so is
ther intersection H \cap K.
(b) Prove that the intersection of an arbitrary nonempty collection of
subgroups of G is again a subgroup of G (do not assume that the
collection is countable). 
Assignment 2:
Assigned 9/23/04
Section 2.2 #10
Section 2.3 #24
Section 2.4 #16

Section 2.2
#10. Let H be a subgroup of order 2 in G. Show that N_G(H)
= C_G(H). Deduce that if N_G(H) = G then H \leq Z(G)
Section 2.3
#24. Let G be a finite group and x \in G
(a) Prove that if g \in N_G(<x>) then g x g^{1} = x^a for some a
\in \Z
(b) Prove conversely that if g x g^{1}  x^a for some a then g \in
N_G(<x>) [Show first that g x^k g^{1} = (g x g^{1})^k = x^{ak}
for any integer k, so that g <x> g^{1} \leq <x>. If
x has order n, show the elements g x^i g^{1} for i=0..n1 are all
distinct, so that g<x>g^{1} = <x> = n and conclude
that g <x> g^{1} = <x>]
Section 2.4 #16. A subgroup M of a group G is called a maximal
subgroup if M \neq G and the only subgroups of G which contain M are M
and G.
(a) Prove that tif H is a proper subgroup of the finite group G then
ghtere is a maximal subgroup of G containing H.
(b) Show that the subgroup of all rotations in a dihedral group is a
maximal subgroup.
(c) Show that if G=<x> is a cyclic group of order n\geq 1
then a subgroup of H is maximal if and only if H = <x^p> for some
prime p dividing n.

Assigned 9/28/04
Section 3.1 #18, 24
Section 3.2 #22

Section 3.2
#18. Let G be the quasidihedral group or order 16 (lattice of
subgroups is on p. 72) G= < s, t  s^8 = t^2 = 1, s t = t s^3 >
and let {\bar G} = G/<s^4> be the quotient of G by the subgroup
generated by s^4 (s^4 is in the center and hence this subgroup is
normal).
(a) Show that the order of {\bar G} is 8
(b) Exhibit each element of {\bar G} in the form {\bar t^a s^b} for
integers a and b
(c) Find the order of each of the elements of {\bar G} exhibited in (b)
(d) Write each of the following elements of {\bar G} in the form from
part (b): {\bar s t}, {\bar t s^{2} t} {\bar t^{1} s^{1} t s}
(e) Prove that {\bar G} \simeq D_8
#24. Prove that if N is a number subgroup of G and H is any subgroup of
G then N \cap H is a normal subgroup of H.
Section 3.2
#22. Use Lagrange's theorem in the multiplicative group to show Euler's
Theorem: a^{\phi(n)} \equiv a (mod n) for all a in {\mathbb Z}
relatively prime to n.

Assigned 9/30/04
Section 3.3 #2

Section 3.3
#2. Prove all parts of the Lattice Isomorphism Theorem

Assignment 3:
Assigned 10/05/2004
Section 3.4 #6,9 
Section 3.4
#6. Prove part (1) of the JordanHölder Theorem by induction on
G.
#9. Prove the following special case of the JordanHölder Theorem:
assume the finite group G has two composition series
1 = N_0 \trainglelefteq N_1 \trainglelefteq \cdots \trainglelefteq N_r
= G and
1 = N_0 \trainglelefteq N_1 \trainglelefteq N_2 = G
Show that r=2 and that the list of composition factors is the same.
(Use the second isomorphism theorem). 
Assigned 10/06/2004
Section 3.5 #17
Section 4.1 #10 a, c, d
Section 4.2 #11
Section 4.3 #12, 13

Section 3.5 #17 If x
and y are 3cycles in S_n, prove that <x, y > is isomorphic to
Z_3, A_4, A_5 or Z_3 x Z_3.
Section 4.1 #10 For x \in G and H, K subgroups of G, the H K double
coset of x in G is H x K = { hxk : h \in H, k \in K}.
(a) Prove that HxK is the union of the left coests x_1 K, x_2 K, ...,
x_n K where { x_1 K, ..., x_n K } is the orbit containing xK of H
acting by left multiplication on the set of left cosets of K.
(c) Show that HxK and HyK are either the same set or are disjoint for
all x, y \in G. Show that the set of HK double cosets partitions
G.
(d) Prove that H xK = K H : H \cap xKx^{1}.
Section 4.2 #11 Let G be a finite group and let \pi : G > S_G be
the left regular representation. Prove that if x is an
element of G of order n and G = mn, then \pi(x) is a product of m
ncycles. Deduce that \pi(x) is an odd permutation if and only if
x is even and G/x is odd.
Section 4.3 #12 Find a representative for each conjugacy class of
elements of order 4 in S_8 and S_12
#13 Find all finite groups which have exactly two conjugacy classes.

Assigned 10/13/04
Section 4.4 #1,20
Section 4.5 #1820

Section 4.4
#1, If \sigma \in Aut(G) and \phi_g is conjugation by g, prove
\sigma \phi_g \sigma^{1} = \phi_{\sigma(g)}. Deduce that Inn(G)
is a normal subgroup of Aut(G).
#20 For any finite group P, let d(P) be the minimum number of
generators of P. Let m(P) be the maximum of he integers d(A) as A
runs over all abelian subgroups of P. Define J(P) = < A  A is
abelian subgroup of P with d(A) = m(P)>
(a) Prove that J(P) is a characteristic subgroup of P.
(b) For each of the following groups of P, list all ableaian subgroups
A of P that satisfy d(A)=m(P): Q_8, D_8, D_16 and QD_16 (where QD_16 is
in section 2.5)
(c) Show that J(Q_8) = Q_8, J(D_8) = D_8, J(D_16) = D_16, J(QD_16) is a
dihedral subgroup of order 8 in QD_16
(d) Prove that if Q is a subgroup of P and J(P) is a subgroup of Q,
then J(P) = J(Q). Deduce that if P is a subgroup (not necessarily
normal) of the finite group G and J(P) is contained in some subgroup Q
of P such that Q is normal in G, then J(P) is normal in G.
Section 5.5 #18 Prove that a group of order 200 has a normal Slow 5
subgroup.
#19 Prove that if G=6545, then G is not simple
#20 Prove that if G=1365, then G is not simple.

Assignment 4:
Assigned 10/21/04
Section 5.4 Let H and K be subgroups of G. Prove if H \cap K =
1 then each element of HK can be written uniquely as a product hk for
some h \in H and k \in K.

Note: On
10/26, I messed up the definition of \semidirect product. Some of
the conclusions that I was drawing from it were incorrect. Please
read your notes from 10/26 and correct them.
The correct HOMEWORK problem should be close to what I stated on 10/21:
If Z_n is a normal subgroup in G and Z_2 a subgroup of G, G is
isomorphic to Z_n \semidirect_\psi Z_2 where \psi : Z_2 > Aut(Z_n)
= Z_{\phi(n)}. Show that if \psi=identity then Z_n
\semidirect_\psi Z_2 = Z_n x Z_2 and if \psi is not the identity, then
Z_n \semidirect_\psi Z_2 is isomorphic to D_{2n}
Assigned 10/26/04
Section 5.5 #12 Classify all groups of order 20 (there are 5
isomorphism types).

Prove that if G is
nilpotent then (H is a proper subgroup of G implies H is a proper
subgroup of N_G(H))
See section 6.1 Theorem 3 (1) => (2), you may prove this the same
way but do not refer to this result.

Assigned Nov 2: If N
is a subgroup of Z(G) then G is isomorphic to G/N x N is easily shown
to be false. Therefore change the problem to be:

Find a finite group
G and a subgroup N of Z(G) such that G/N x N is not isomorphic to G.

Nov 4:
Section 6.1 #3: If G is finite prove that G is nilpotent if and only if
it has a normal subgroup of each order dividing G, and is cyclic if
and only if it has a unique subgroup of each order dividing G.
Section 6.2 #9: Prove there are no simple subgroups of order 336.

Assignment 5:
Nov. 9:
Section 6.2
#12. Prove that there are no simple subgroups of order 9555
#15. Classify the groups of order 105
Also answer: What is R[x] (in the sense of what category does it fall
in) for R in the following categories
1. noncommrings w/1(not a div ring)
2. noncommdivrings
3. fields
4. integral domains
5. commrings w/1
6. commrings without 1
7. noncommrings without 1
Note: this is somewhat imprecise and you may not be able to answer it
for every one of these categories, but try as much as you can. I
am looking for as detailed an answer as possible.

Nov 11.
Prove one of the (2nd, 3rd or 4th) isomorphism theorems for rings, your
choice.

Assignment 6:
Jan 12.
1,2 from the section on Eisenstein's criterion

Jan 20.
9 problems Groebner bases. Include
annotated printout explaining the answer.

Assignment 7:
Feb 24.
p. 344 #8 An element m of the RModule is called a torsion elment if
rm=0 for some nonzero element r \in R
(a) if R integral domain then Tor(M) is a submodule
(b) Give an example of an integral domain and a module where Tor(M) is
not a submodule (consider M=R)
(c) If R has zero divisors then show that every nonzero Rmodule has
nonzero torsion elements.
p. 356 #11 Show that if M_1&M_2 are irrducible Rmodules then any
nonzero Rmodules homomophism from M_1 to M_2 is an isomorphism.
Decduce that if M is irreducible then End_R(M) is a division ring.
Show explicitly that Q\otimes Z^n \cong Q^n
p. 4034 #6 Prove that TFAE for a ring R: (i) every R module is
projective (ii) Every Rmodule is injective.
#7 Let A be a nonzero finite abelian group: (a) prove A is not a
projective Zmodule (b) prove that A is not an injective Zmodule

Assignment 8:
March 3:
Consider Q( a+b sqrt(d) ), b,d \neq 0
(1) Calculate A = M_{1,sqrt(d)}^{1,sqrt(d)}(\psi) \in M_{2x2}(Q)
where \psi: Q(sqrt(d)) \rightarrow Q(sqrt(d))
\psi(r) = r(a+b sqrt(d))
(2) Find p(x), the characteristic polynomial of A
(3) Show that p(a + b sqrt(d)) = 0. Find the other root of p(x),
\beta. Why do you know that Q(a+b sqrt(d)) \cong Q(\beta)
(4) Show explicitly that { a_0 I + a_1 A : a_0, a_1 \in Q } \cong Q(a+b
sqrt(d)) as Qalgebras
(5) We certainly have Q(a+b sqrt(d)) is contained in Q(sqrt(d)).
Does the reverse inclusion hold?

March 10:
p 545, #3 Determine the splitting field and its degree over Q of x^64

March 15:
p 545, Determine the splitting field and the lattice corresponding lattice of subfields
of x^4+2 and x^4+x^2+1

March 22:
Read page 599600: Find an n such that Gal( Q(\zeta_n)/Q ) = Z_4 x Z_2 x Z_9 x Z_3


