Syllabus This semester's course will follow the Artin text fairly closely.
Groups and vector spaces
Group axioms, definitions, examples (Artin Ch 2)
Vector spaces (Ch 3)
Linear transformations (Ch 4)
Properties of groups (Ch 6)
Rings, fields, and solving equations
Ring theory (Ch 10)
Field theory, ruler-and-compass constructions (Ch 13)
Galois theory, solving equations by radicals(Ch 14)
Midterm Test I: In class Wed Mar 1. Come 5 min early for extra time. Test questions will emphasize theory more than Quiz questions do, so review both Quiz and HW problems. Topics:
Abstract group theory
Ch 2.1 Definitions & examples
Ch 2.2 Subgroups, cyclic groups
Ch 2.3 Isomorphisms of groups
Ch 2.8 Product groups
Ch 2.4--2.6, 2.9--2.10: Cosets of a subgroup, quotient group Homomorphisms φ : G → H , Im(φ) ≈ G / Ker(φ) .
Ch 3.4 Coordinates of a vector with respect to a basis [v]B , isomorphism V ≈ Fn Change of basis matrix CBA [v]A = [v]B
Ch 3.6 Sum V ⊕ W
Quotient V/W , same as quotient in group (V , +)
Ch 4.1 Linear mapping φ : V → W Im(φ) ≈ V / Ker(φ) , rank(φ) = dim(V) - nullity(φ)
Ch 4.2 Matrix of a linear mapping , [φ]BA [v]A= [φ(v)]B
Old Homework: I will not collect homework (except problems marked with ♠), but it will be the basis of the weekly quizzes (usually on Fridays). You may hand in Extra Credit problems at any time during the course, no matter when they were posted. Exercise and page numbers refer to the Artin text.
HW 1/13 Homework instead of Quiz: group theory review from 418H. Notes: 11/31 , 12/9. Reference: Ch 2.5, 2.6, 2.9, 2.10. Hand in the problem marked with ♠ below.
♠ The quaternion group consists of the 8 matrices listed on p. 48 under the operation of matrix multiplication: Q = {±1, ±i, ±j, ±k}. We also know the following groups of order 8: the dihedral group D4 , and the abelian groups C8 , C4×C2 , C2×C2×C2 .
Prove that none of these five groups are isomorphic to each other. Hint: Find unique features of the multiplication table of each group G, looking especially at the order of each element g ∈ G.
Prove that any group of order 8 is isomorphic to one of these.Hint: Consider separately the cases in which the maximum order of an element is 8, 4, and 2. Recall that a subgroup of index 2 must be normal, and that the conjugate xgx-1 has the same order as g.
Ex #2.1.10, p 69: algebra in a group.
Ex #2.2.17, p 70: algebra in a group.
Ex #2.3.1, p 71: a surprising isomorphism.
Ex #2.4.10, p 72: kernel of a homomorphism.
Ex #2.6.1, p 74: index of a subgroup.
Ex #2.6.2, p 74: cosets do not overlap.
The order of a group means the number of elements in it, whereas the order of an element g ∈ G is the smallest k > 0 with gk = 1 .
Prove that gk = 1 if and only if ord(g) | k (that is, the order of g divides k). Hint: Use Lagrange's Theoremthat |H| divides |G| for any subgroup H ⊂ G.
Show that for any group G of order n, we have gn = 1 for any g ∈ G . What does this mean about the order of the element g ? (Not just ord(g) < n.)
Let (Zp , + , ×) be the field of clock arithmetic modulo a prime p, and let (Zp× , ×) denote the group of non-zero elements of Zp under the multiplication operation.
Show that xp-1 = 1 for any x ∈ Zp×.
Fermat's Little Theorem: For any integer n and any prime p, we have p | np - n . Example: 25 - 2 = 30 , a multiple of 5. Prove the theorem using the above group properties of Zp×.
Wilson's Theorem: For any prime p, we have p | (p-1)! + 1 . Example: (5-1)! + 1 = 25 , a multiple of 5. Prove the theorem using the above group properties of Zp×.
H1={ x | x•v1= x•v2 = 0 } H2={ x = λ1v1+λ2v2 | λ1,λ2 ∈ R }
Here H1 is defined by dot-product equations, whereas H2 is parametrized by (λ1,λ2).(These are the dual forms for writing any k-dimensional plane in Rn.) Write each of these planes in the other form.Hint: First solve a system of linear equations to find two vectors orthogonal to both v1 and v2 . (b) Now let v0 = (1,1,1,1) , and define affine planes (planes not containing the origin) by:
H3={ x | x•v1= x•v2 = 1 } H4={ x = v0+λ1v1+λ2v2 | λ1,λ2 ∈ R }
Write each of these affine planes in the other form. (c) Write both of the above forms for the unique 2-dimensional plane H5 which contains the three points v0 , v1 , v2 . Hint: Find two directions which point along H5 .
Use row reduction to solve the system of equations (2.13) on Artin p. 13. Compare your solution to that at the top of p. 13. Notice that in this case, one cannot get the identity matrix located in the columns 1,2,3, but rather in columns 1,2,4. Thus, in writing the solution vectors, variables 1,2,4 will be the dependent ones.
Artin Exercises 3.1.1--3.1.7, p. 104.
Ex 3.3.1, p. 105: Find a basis for W = Span(w1,...,w4) ⊂ R4 . Hint: these vectors are linearly dependent. Perform column reduction to simplify the spanning set.
Ex 3.3.2, p. 105: Use row reduction.
Suppose we have a solid P ⊂ Rn defined by the inequalities:
For example, a triangle T with vertices (0,0), (1,0), (0,1) has equations:
0 ≤ x1 ≤ 1 0 ≤ x2 ≤ 1 - x1
and 2-dimensional volume (area):
∫01∫01-x1 ( 1 ) dx2 dx1= ∫01 (1 - x1) dx1= 1 / 2
Problem: Find the n-dimensional volume of the standard simplex with vertices (0,...,0) , (1,0,...,0) , .... , (0,...,0,1).
HW 2/3 Quiz: Repeat of last week's quiz on subspaces & reduction. Reference for this week: Ch 3.3.
Ex 3.3.3, p. 105.
Ex 3.3.5, p. 106.
Ex 3.3.7, p. 106. Show that in the vector space of real functions, the elements x3 , sin(x) , cos(x) are linearly independent. Hint: Fix 3 points x1 , x2 , x3 , and for each function f, consider the triple:
v(f) := ( f(x1) , f(x2) , f(x3) ) ∈ R3 .
Show that if v(f1) , v(f2) , v(f3) are linearly independent, then so are the three functions.
Carefully prove the equivalence of the three definitions of linear independence. That is, given a vector space V over a field F, and a set L = {v1 , ... , vn}, the following are equivalent:
Any vector v has at most one representation as v = c1v1 + ... + cnvn .
The only linear relation c1v1 + ... + cnvn = 0 is given by c1 = ... = cn = 0 .
None of the vectors can be written in terms of the others: there is no expression vi =∑j ≠ i cjvj .
HW 2/10 Quiz: Bases, change of coordinates. Reference: Ch 3.4.
Ex 3.4.1--3.4.4, p. 106.
Ex 3.4.5, p. 106: Prove that a given set of vectors is a basis of R3. Find the coordinates of a given vector with respect to the basis. When does this work modulo p?
HW 2/13 Hand in the problem below (same as Prob 4 from HW 2/3).
♠Carefully prove the equivalence of the three definitions of linear independence. That is, given a vector space V over a field F, and a set L = {v1 , ... , vn}, the following are equivalent:
Any vector v is a linear combination of the vi's in at most one way: v = c1v1 + ... + cnvn has at most one solution for (c1,...,cn).
The only linear relation c1v1 + ... + cnvn = 0 is given by c1 = ... = cn = 0 .
None of the vectors is a linear combination of the others: there is no expression vi =∑j ≠ i cjvj .
Hint: Prove each implication (a)⇒(b) , (b)⇒(a) , (b)⇒(c) , (c)⇒(b) , stating the hypothesis in each case.
HW 2/17 Quiz: Properties of abstract vector spaces and linear mappings (e.g., linear independence, image, kernel).
Prove: If φ : V → W is a linear mapping, and the set { φ(v1), ... , φ(vk) } is linearly independent in W, then { v1, ... , vk } is linearly independent in V. Does the same hold with "linearly independent" replaced by "spanning"?
Recall that we defined the direct sum of vector spaces as:
V ⊕ W = { (v,w) | v∈V , w∈W }
with componentwise addition and scalar multiplication.
Suppose V has basis {v1,...,vn} , and W has basis {w1,...,wm}. Show that V⊕W has basis
{ (v1,0), ... , (vn,0) , (0,w1) , ... , (0,wm) }.
That is, show that these elements are linearly independent and span.
Show that the set { (vi,wj) , i=1,..,n , j=1,..,m } is not linearly independent, nor spanning, even if dim(V) = dim(W) = 2, by finding a linear relation. Hint: For dimension 2, write the elements (vi,wj) in the coordinates of the above basis, getting a 4 × 4 matrix. To find linear relations, row-reduce to solve for the 4 coefficientscij such that ∑ij cij (vi,wj) = 0 . To find the span, column-reduce.
Ex 4.1.1, p. 145: compute Ker and Im of a matrix.
Ex 4.1.3, p. 145: Ker and Im are subspaces.
Suppose V is a vector space over a finite field Zp, so we can consider (V,+) as an abelian group. Write (V,+) as a product of finite cyclic groups Cn.
Consider the space of polynomials R[x] as an (infinite-dimensional) real vector space. Determine the image and kernel of the linear mappingsφ,ψ : R[x] → R[x] defined by
φ(f(x)) := x f(x) and ψ(f(x)) := (f(x)-f(0))/x .
Can you get results like this for a finite dimensional vector space?
HW 2/24 Quiz: Normal subgroups, quotient groups. Reference: Ch 2.10, pp. 66-69. Theorem:
Given a normal subgroup N ⊂ G, we have a surjective morphism
φN : G → G/N g → gN
with Im(φN) = G/N and Ker(φN) = N .
Given a group morphism φ : G → H, we have an isomorphism:
φ : G/N ≈ Im(φ) gN → φ(g)
Exercises:
Consider the map
φ : (R , +) → (C×, •) x → z = cos(x) + i sin(x) .
Show that φ is a group morphism. Find Im(φ) and Ker(φ), and verify the isomorphism ψ : R / Ker(φ) ≈ Im(φ) .
Ex 2.10.9, p. 76: Quotient of groups undoes product of groups.
Ex 2.10.10, p. 76: Quotents of the multiplicative group of complex numbers C×. Also, prove that
(r,θ) → z = r cisθ
gives an isomorphism R>0 × R/2πZ ≈ C×
Show that x → ex is an isomorphism from (R , + ) to (R>0 , •). Thus :
(x,θ) → z = ex cisθ
gives an isomorphism R × R/2πZ ≈ C× .
Ex 2.10.4, p.76: Cosets in 6-element groups.
Ex 2.6.12, p.74. Cosets in a matrix group.
Ex 2.7.5 & 2.7.6, p.75: Product of a normal and non-normal subgroup.
Test 1 Review: Group theory. The exercises below about conjugacy classes (Ch 6.1, pp. 197-200) also serve as a review of Ch 2. For a group element x ∈ G, we defined the centralizer to be the elements g which conjugate x trivially:
Z(x) := { g ∈ G s.t. gxg-1 = x } .
The conjugacy class of x is the set of all elements conjugate to x:
K(x) := { gxg-1 s.t. g ∈ G } .
We showed that gxg-1 = hxh-1 if and only if g = h mod Z(x), so that each coset g Z(x) ∈ G/Z(x) corresponds to just one conjugate of x.Thus:
|K(x)| = |G| / |Z(x)| .
The class equation is:
|G| = ∑ |G| / |Z(xi)|
where xi runs over a set of representatives for the conjugacy classes K(x1), K(x2), ....
Let G = D3, generated by x,y subject to x3 = y2 = 1 and xy = yx-1. Verify that the conjugacy classes are: K1 = {1} , K2 = {x,x2} , K3 = {y,yx,yx2}. For each class K, choose x ∈ K and determine Z(x). Find the cosets G/Z(x), and verify that each one corresponds to just one conjugate of x, and that |K(x)| = |G/Z(x)| = |G| / |Z(x)| .
Ex 6.1.8, p.229. For each of the following groups G, determine its partition into conjugacy classes. For each class K(x), compute Z(x) and verify that |K(x)| = |G| / |Z(x)| : G = D4 , D6 , Dn , quaternion group Q ; the 12-element group GL2(Z3) of 2×2 invertible upper triangular matrices with entries in Z3 :
Classify the groups of order 21 = (3)(7) by using the Sylow Theorems for p = 3 and p = 7 . Show that there are 2 isomorphism types: the cyclic group C21 , and one non-abelian group whose multiplication is defined by: x7 = y3 = 1 and one other relation.
Ex 6.4.2 & 6.4.3, p.231: for p,q distinct primes, any group with |G| = pq has a non-trivial normal subgroup. Same for |G| = p²q . (We say G is simple if it is only normal subgroups are the trivial ones N = {1} and N = G.Such a G is "prime" since it has no non-trivial quotient groups.)
Recall the definition of a group G acting on a set S. Our main examples are:
The left-multiplication action of G on S = G itself by g*s = gs (multiply in the group).
The conjugation action of G on S = G itself by g*s = gsg-1 .
The secondary actions derived from each of the above primary actions: G acting on
S = { all subsets S ⊂ G }
by g*S := { g*s for s ∈ S }.
Our proofs of the Sylow Theorems for a group with |G| = pe m depend on variations of these actions. Problem: Consider the dihedral group G = D3 , the symmetry group of the triangle. Follow the proofs of the Sylow Theorems for this special case and p = 2 and p = 3, by explicitly examining:
G acting by left multiplication on S = { all subsets of size pe } . Find an orbit OS whose order is prime to p. Then H = StabG(S) is a Sylow p-subgroup.
Given a subgroup K ⊂ G and a Sylow p-subgroup H ⊂ G, consider K acting by left multiplication on S = { gH for g ∈ G } = G/H, the set of H-cosets of G. Consider an orbit OgH whose order is prime to p, and see that gHg-1 ∪ K ⊂ K is a Sylow p-subgroup of K.
G acting by conjugation on S = { all Sylow p-subgroups of G } . Then S is a single orbit OH, where H ∈ S. Then #S = #G / #NG(H) divides #G / #H = m.
A particular Sylow p-subgroup H acting by conjugation on S = { all Sylow p-subgroups of G }. Then all orbits have order divisible by p except for OH = {H}. Thus #S ≡ 1 mod p.
HW 3/24 No quiz this week. Reference: Ch 6.3, 6.4, pp. 203--208.
Abelian groups of order 21.
Prove: Let G be a group with normal subgroups H and K, such that G = H.K := { hk s.t. h∈H , k∈K} and H ∩ K = {1}. Then hk = kh for all h∈H , k∈K , and G ≈ H × K .
Use the above result to show that if |G| = 21 and G has a unique Sylow 3-subgroup, then G ≈ C3 × C7 ≈ C21 .
HW 3/31 (Part 1) Hand in the problem marked with ♠ . Reference: Ch 6.7, 6.8. Definitions
The Cayley graph of a group with n generators, G = < g1,...,gn > , is a network with the elements of G as vertices, which are connected by certain arrows, each having a label j = 1,...,n . Namely, for each g and j we have the outgoing j-arrow g →j g gj ,and the incoming j-arrow g gi-1 →j g .
The free group on n generators:
F(n) = F(a1,...,an)
is the set of all words in the 2n-letter alphabet
{a1,...,an,a1-1,...,an-1} ,
including the empty word with no letters which we denote as 1. We can reduce words via the relations aj aj-1 = 1 = aj-1aj , but otherwise all words are distinct. We multiply words by concatenating.
Given a set of words r1,...,rm ∈ F(n), we define the finitely presented groupgenerated by aj with relations rk as the quotient group:
where N(r1,...,rm) means the smallest normal subgroup of F(n) containing the rk's: that is, the group of all products of words wrkw-1 and wrk-1w-1 for w ∈ F(n).
Proposition: We can produce the Cayley graph of a finitely presented group by drawing the infinite tree Cayle graph of F(n), and progressively identifying the beginning and endpoint of each path corresponding to a relation rk . If such an idenitfied graph has a closed path corresponding to each relations at each point, it is the Cayley graph of the fintely presented group. Problems
List all reduced words of 3 letters in F(2) = F(a,b).
Let w, w' ∈ F(n) be words. We can express the relation w = w' by r1 = w(w')-1 = 1 orr2 = (w')-1w = 1 or r3 = w'w-1 = 1 or r4 = w-1w' = 1 . Show that these are all equivalent:
We saw that the only possible non-abelian group G of order 21 is given by generators < x , y > and the relations (*) above.Prove that there really is such a group by constructing its 21-vertex Cayley graph. Start with a 7-cycle of x-edges corresponding to , then add 3-cycles of y-edges at each vertex: {g, gy, gy2}. Finally, add x-edges corresponding to x = y-1x2yand to x = yx4y-1 . Show that this really is the Cayley graph of the finitely presented group by finding the appropriate closed paths in the graph. Identify the unique Sylow 7-subgroup and the seven Sylow 3-subgroups.
Ex 6.4.4, p. 231. The group G defined in the exercise is a matrix construction of the above group. Show that it really has order 21, and find generators x,y ∈ G which satisfy the relations given above.