Course Description

Course Description

Homework, solutions, handouts

Math 6120- Modern Algebra


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 k-tuples of the numbers 1 through is not clear if they are allowing the entries in the k-tuples 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 k-element subsets is faithful
(b) for which values of k the action of S_n on k-tuples 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 non-empty 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..n-1 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 quasi-dihedral 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 Jordan-Hölder Theorem by induction on |G|.
#9. Prove the following special case of the Jordan-Hölder 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 3-cycles 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 n-cycles.  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 #18-20
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. non-comm-rings w/1(not a div ring)
2. non-comm-div-rings
3. fields
4. integral domains
5. comm-rings w/1
6. comm-rings without 1
7. non-comm-rings 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 R-Module 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 R-module has nonzero torsion elements.
p. 356 #11 Show that if M_1&M_2 are irrducible R-modules then any nonzero R-modules 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. 403-4 #6 Prove that TFAE for a ring R: (i) every R module is projective (ii) Every R-module is injective.
#7  Let A be a nonzero finite abelian group: (a) prove A is not a projective Z-module (b) prove that A is not an injective Z-module

Assignment 8:
March 1:
Assignment on vector spaces
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 Q-algebras
(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^6-4

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 599-600: Find an n such that Gal( Q(\zeta_n)/Q ) = Z_4 x Z_2 x Z_9 x Z_3