MTH 418H
Honors Algebra I

1. Integers Z, Euclidean algorithm for gcd, primes, unique factorization, irrationals
   Ring axioms, polynomial rings, factorization of polynomials, linear factors, Euclidean rings
2. Rational numbers Q, field axioms, fraction fields
3. Real numbers R, construction, defining axioms of complete ordered field
   Intermediate Value Theorem
4. Complex numbers C, construction, algebra of plane vectors, analytic functions
   Line integrals, Cauchy Formulas, Fundamental Theorem of Algebra

1. Group axioms, definitions, examples (Artin Ch 2)
2. Vector spaces (Ch 3)
3. Linear transformations (Ch 4)
4. Symmetry (Ch 5)
5. Properties of groups (Ch 6)

Also look at the Definitions Sheets 1, 2, 3 (new!). Review your quizzes thoroughly, but be aware that the Test questions will be somewhat harder. I will be happy to discuss any questions in my office hours, and I will also hold a review lecture Wed Oct 13.

1. Use the following method to find all the prime numbers less than 100. Write a list of all the numbers 2,3,4,...,100, circle the first number (which is 2) and cross out all multiples of 2. Circle the first uncrossed number (which is 3) and cross out all multiples of it. Repeat until all further numbers are crossed out. The circled numbers are the primes. You can use a print-out of this table.
2. Let π(x) be the number of primes p ≤ x . The amazing Prime Number Theorem (which we won't prove in this course) states that π(x) tends toward x / log(x) as x→∞, in the sense that the percentage error tends to zero. (Here log means natural logarithm, base e.) Make a graph or table of the functions π(x) and x / log(x) for x ≤ 100 , and verify that the percentage difference between the two gets smaller.
3. What does the Prime Number Theorem say about the density of prime numbers in a large interval {1,2,...,n}?

1. Find the polynomial gcd( f(x), g(x) ) .
2. Write   gcd( f(x), g(x) ) = m(x)f(x) + n(x)g(x)   for some m(x), n(x).
3. Factor f(x) and g(x) into irredicible polynomials in Q[x](the analog of prime numbers).

1. Show that for an integer k, the n^th root k^1/n is rational if and only if k is a perfect n^th power, i.e., k = mⁿ for some integer m.Hint: Apply the Rational Root Test to f(x) = xⁿ-k
2. If n is a natural number and p a prime, show that   f(x) = xⁿ - p   is irredicible in Q[x] . Use the Eisenstein Criterion.
3. If p is a prime, show that   f(x) = x^p-1 + x^p-2 + ... + x + 1   is irreducible in Q[x] . Hint: f(x) is irreducible if and only if f(x+1) is. (Why?) Now,  f(x) = (x^p-1) / (x-1) , so  f(x+1) = x ((x+1)^p -1) . Use the Binomial Theorem and apply Eisenstein.
4. Hand in for Extra Credit: What can you say about the irreducible factors of   xⁿ - 1   for arbitrary n ? Hint: There is an irreducible factor Φ_d(x) for each divisor d | n , called the d^th cyclotomic polynomial. For example, x⁶-1 = (x³-1)(x³+1) = (x-1)(x²+x+1)(x+1)(x²-x+1) ,where Φ₁(x) = x - 1 , Φ₂(x) = x + 1 , Φ₃(x) = x² + x + 1 , Φ₆(x) = x² - x + 1 .Can you predict Φ_d(x) for various d ? Can you prove your polynomials are irreducible? Can you give a formula for the degree of Φ_d(x)?

1. Show that a value v is in an error-window [c-ε, c+ε]   if and only if   |v-c| ≤ ε . Reformulate the definition of continuity in this notation.
2. Triangle inequality: Show that |a+b| ≤ |a| + |b| for real numbers a,b.
3. Prove that f(x) = x and f(x) = const are continuous functions, by prescribing a δ for each ε.
4. Prove that if f(x), g(x) are functions continuous at x = a, then f(x) + g(x) and f(x)*g(x) are also continuous at x = a, and so is 1/g(x) if g(a) ≠ 0.
   Hint: The first should go like this. Given f(x), g(x) continous at x = a, and an error tolerance ε>0, take δ small enough that |x-a| ≤ δ guarantees |f(x)-f(a)| ≤ ε/2 and |g(x)-g(a)| ≤ ε/2 . This is possible because we can meet the y-error tolerance ε/2 for f(x) and for g(x) by choosing sufficiently small x-error tolerances δ₁ and δ₂ : let our δ be the minimum of these two. Now |x-a| ≤ δ guarantees:|f(x)+g(x)-f(a)-g(a)|  = |f(x)-f(a)+g(x)-g(a)|  ≤  |f(x)-f(a)| + |g(x)-g(a)|  ≤  ε/2 + ε/2 = εThe other two are harder.

1. Show that {s < 0} is the additive identity.
   
2. Define the product and reciprocal of real numbers in terms of sets, and justify your answer. (See the Spivak handout for answers.)
   

1. Really a real number: that is, satisfies the definition in the previous problem;
2. An upper bound of A ;
3. Less than or equal to any other upper bound of A .

1. Show that every real quadratic equation, ax² + bx + c = 0 with a,b,c ∈ R , has a solution in C .
2. Show that every complex number a+bi has a complex square root. Hint: Write out (c+di)² = a+bi , set the real and imaginary parts equal on the two sides, and solve the system for the real numbers c,d.
3. Show that every complex quadratic equation ( a,b,c ∈ C ), has a solution in C .

1. Add and multiply two matrices of the form

   a	-b
   b	a

   to verify that they form a copy of the complex number field C inside the ring of 2 × 2 matrices with real entries, M_2×2(R) . Formally, this means that if we define the map φ : C → M_2×2(R) by setting φ(a+bi) equal to the above matrix, then φ is a morphism of rings. Note that M_2×2(R) is not a commutative ring nor a division ring, but the subset consisting of the above matrices is actually a field, isomorphic to C .
2. Show that the complex conjugate corresponds to transpose of matrices, and the norm or length of a complex vector is the square root of the determinant of the corresponding matrix.
3. Extra Credit: Let us try to form a "hypercomplex" number system consisting of matrices of the above form with complex number entries a,b ∈ C . Show that these matrices form a commutative ring but not a field.
4. Extra Credit: On the other hand, consider complex matrices of the form:

   a	-b
   b	a

   where a,b ∈ C and a means the complex conjugate. Show that these matrices form a division ring but not a field. This ring is called the quaternions, usually denoted H, a fascinating number system which relates to three dimensions much as complex numbers relate to two dimensions. I can suggest you more exercises on them if you are interested.

1. Show that if α = r cis(θ), where r = ||α|| and θ = arg(α), then the n complex n^th roots of α are r^1/n cis(θ/n) ζ_n^k for k = 0,1,...,n-1 .
2. Use the half-angle formulas in trigonometry to compute √α = c + di for any α = a + bi , re-doing one of the above problems.
3. Extra Credit: Show that the cyclotomic polynomial Φ_n(x) is the product of (x - ζ_n^k) for all positive k < n with gcd(k,n) = 1. How many such k are there (namely, what is the degree of Φ_n(x) ) ?

1. The matrix of partial derivatives

   ux	uy
   vx	vy

   is a complex multiplication matrix, satisfying the Cauchy-Riemann equations v_x = -u_y and v_y = u_x .
2. The gradient vector fields ∇u = (u_x, u_y) and ∇v = (v_x, v_y) are 90° rotations of each other. Thus, the level curves u(x,y) = const and v(x,y) = const are perpendicular to each other at their points of intersection.
3. There is a well-defined complex number given byf '(a+ib) = lim_ε→0( f(a+ib+ε) - f(a+ib) ) / ε= lim_ε→0( f(a+ib+iε) - f(a+ib) ) / iε

1. Explain why this measures the strength of the net counterclockwise rotation of the flow associated to f around the curve C. Hint: the dot product is the product of the length of two vectors times the cosine of the angle between them.
   
2. If C is parametrized by a function r(t) = (x(t),y(t)) for 0≤t≤1, we can compute circulation as an ordinary integral:∫_C f(r)•dr = ∫₀¹ f(r(t)) •r'(t) dt
   = ∫₀¹ u(x(t),y(t)) x'(t) + v(x(t),y(t)) y'(t) dt .Compute these integrals over a circle C with radius r and center 0 for the vector fields given by f(z) = const, f(z) = z, and f(z) = iz, and verify that, as one would expect from the pictures, the first and second have no circulation, while the third has positive circulation proportional to the radius r.
   
3. Repeat the above exercises for the flux defined as: ∫_C f(r)×dr := lim_n→∞∑_i=1ⁿ  f(r_i) × (r_i - r_i-1) .Show that this measures the net outflow of f from the region surrounded by C . (Hint: the cross product of two vectors is the area of the parallelogram that they span.) You should find that f(z) = const and f(z) = iz have no flux, and f(z) = z has positive flux.
   
4. Finally, consider f(z) as a complex function, and parametrize the curve C by a function z(t) = x(t) + iy(t) , with sample points z_i . The complex line integral defined by:∫_C f(z)*dz := lim_n→∞ ∑_i=1ⁿ  f(z_i) * (z_i - z_i-1) = ∫₀¹ f(z(t)) *z'(t) dt ,where * indicates multiplication of complex numbers, and z'(t) = x'(t) + iy'(t).The previous two integrals take real values, but the line integral gives a complex number. Show that the real and imaginary parts are:∫_C f(z)*dz =∫_C f(z)•dz  + i ∫_C f(z)×dz ,the circulation and flux integrals for the conjugate vector field f(z) = u(x,y) - i v(x,y) . Hint: Use Problem 4 above.Compute this integral for the three functions f(z) above. Hint: The answers should all be 0. That is, f(z) has zero circulation and divergence for each f(z). Cauchy's Integral Formula asserts that this is true for any function which is complex analytic at every point of the region enclosed by C.

1. In class we defined the groups S_n = {permutations of an n-set}, and D_n = {symmetries of a regular n-gon}.
   1. Compute the multiplication table of S₃ using the two-line notation for permutations.
   2. Compute the multiplication table of D₃ = S₃ using as generators a rotation ρ and a reflection α, along with the relations ρ³ = α² = 1 and ρα = αρ² .
   3. Compute the multiplication table of D₄ by finding a convenient set of generators, and verifying their relations via the permutation notation. Hint: Very similar to D₃.
   4. Extra Credit: How many elements are there in D_n for an arbitrary n? Can you give a good set of generators and relations?
2. Prove that if a,b are elements of a group, then (ab)^-1 = b^-1 a^-1 .
3. Ex #2.1.5, p. 69. Algebra in a group.
4. Ex #2.1.10, p. 69. Algebra in a group.

1. Ex #2.2.6 and 2.2.8, p. 70 (see p.48). Two more examples of groups.
2. Ex #2.2.10 and 2.2.11, p. 70. Order of an element.
3. Ex #2.2.2, p. 70. Algebra in a group.
4. Ex #2.2.17, p. 70. Algebra in a group.
5. Ex #2.3.1, p. 71. A surprising isomorphism.
6. Ex #2.3.5, p. 71. Definition of isomorphism.
7. Ex #2.3.12, p. 71. The inverse map anti-automorphism.
8. Extra Credit: Ex #2.3.14, p. 71. Computing the group of automorphisms Aut(G) for a group G.
9. Ex #2.4.7, p. 72. A homomorphism.
10. Ex #2.4.10, p. 72. Kernel of a homomorphism.
11. Ex #2.4.13, p. 72. Conjugate and normal subgroups.
12. Extra Credit: Ex #2.4.19, p. 73. A slightly tricky problem.
13. In multiplying elements of the dihedral group D_n, most people had trouble with representing the elements in a precise way. One way is to write a group element as a permutation of the n vertices.
    1. Here is another way, using complex numbers. Let α : C → C be the map α(a+bi) = a-bi , the complex conjugate map. Let ρ : C → C be the map ρ(z) = ζ z , where ζ = cos(2π/n) + i sin(2π/n), an n^th root of unity.
       Geometrically, α is a reflection and ρ a rotation, both symmetries of an n-gon with center at the origin and one vertex at z = 1 .Now use complex-number algebra to compute: α² = 1 ( meaning α(α(z)) = z for all z ), ρⁿ = 1 , and αρ = ρ^-1α .
    2. Here is another way, using matrices. The maps α and ρ are linear mappings of the plane R², so they correspond to 2 × 2 real matrices. Write these down and check the relations between α and ρ using matrix multiplication. (Note: For ρⁿ = 1, you will have to use the angle-sum trig identities, which essentially say that rotation by θ₁ composed with rotation by θ₂ is the same as rotation by θ₁ + θ₂.)
    3. Recall that the 8-element group of quaternion units is Q = { ±1, ±i, ±j, ±k } with relations i² = j² = k² = -1 and ij = k , ji = -k , jk = i , kj = -i , ki = j , ik = -j .Check these relations using the representation of Q by complex matrices on p.48.
    4. Extra Credit: Use the matrix representation of Q to find a 4-dimensional polyhedron P in R⁴ whose symmetry group is Q. Namely, take some random point p = (a,b) in C² , and take 8 points gp for g ∈Q (that is, multiply the matrix g by the column vector p). Consider these 8 points as points in R⁴ by taking real and imaginary parts of the complex coordinates, and take these 8 points as the vertices of your polyhedron. The rest is gotten by taking the convex hull of these 8 points p₁,...,p₈ : that is, all points v = c₁p₁ + ... + c₈p₈ , where the coefficients run over all c₁,...,c₁ ≥ 0 with c₁ +... + c₁ = 1 .
       Figure out the edges of this polyhedron, which are certain "outside" lines between these points (i.e., those lines not in the interior of the 4-dim polyhedron). Also find the 2-dim faces, which are certain polygons with vertices among p₁,...,p₈, lying in 2-dimensional subspaces of R⁴, and not in the interior of the 4-dim polyhedron. Similarly for the 3-dim faces.
14. In class we listed the 5 groups of order 8: the dihedral D₄ , the quaternion Q , the cyclic C₈ , and the product groups C₄ × C₂ ,   C₂ × C₂ × C₂ . For each group, compute the orders of the elements, the centers and the conjugacy classes, and find all normal subgroups.
    Use this intrinsic structural data to prove that these 5 are all non-isomorphic: that is, there is no possible isomorphism between them.
    Note: Some of my computations of these in class were too hasty, so don't trust them too much.

• Ex #4.10, 4.16, 4.17
• #4.20. SL(n,R) is the group of n x n real matrices with determinant = 1, with the operation of matrix multiplication.
• #4.22, 4.23
• #6.1. The index of a subgroup H in G is the number of distinct cosets gH.
• #6.2, 6.10
• #6.11. We now know enough group structure theory to prove our first classification result: classify all groups of order 6.
• #8.3, 8.4
• #9.2, 9.4
• #10.1. The First Isomorphism Theorem says that if φ: G → G' is an onto homomorphism with kernel H,then G' is isomorphic to G/H.
• #10.4(a), 10.5, 10.9

1. Work out for yourself in detail the definition and properties of the quotient group G/H, whose elements are cosets gH, and whose multiplication is: gH * g'H = {g h1 g' h2 for h1,h2 ∈ H} .Show that the right side is a coset, and that the group axioms hold.
2. Let D₄ be the dihedral group generated by a reflection alpha and a rotation rho of order 4. Show that the cyclic group <ρ²> is a normal subgroup. Determine the multiplication table for the quotient group D₄/<ρ²>, and if it is isomorphic to a group we have seen before.
3. Extra Credit: In representation theory terms, if G is the symmetries of an object X, then a subgroup H is the symmetries of a decorated object X~, and a normal subgroup is the symmetries of an X~ which can be defined without labelling X. Now, can you give a representation theory description of the center Z(G), which is a special normal subgroup? I don't know an answer, or even if there is a good answer.

1. The order of a group is its number of elements #G.Prove: A group G of prime order p is cyclic: G = C_p .Thus, there is only one possibility for a group of order 17, say.
2. Reference: Artin Ch 6.6. The symmetric group S_n is the symmetries of the set [n] = {1,2,...,n}, i.e., all permutations π : [n] → [n] . We can denote a permutation π using the one-line notation [π(1),π(2),...,π(n)] or the 2-line notation, but the most economical is the cycle notation. A cycle of π is a list of the form (i, π(i), π(π(i)),...), which stops when it comes back to the original i. Each permutation can be written as a product of disjoint cycles in several ways, for example: π = [25341] = (125)(3)(4) = (251)(3)(4) = (3)(4)(512)since the cycles are: π(1) = 2 , π(2) = 5 , π(5) = 1 ; andπ(3) = 3 ; and π(4) = 4 .
   1. Show that for any permutation π the conjugacy class C_π := {gπg^-1 | g ∈ S_n} is precisely the set of all permutations with cycles of the same length as π .
   2. Now let n = 4. We can realize S₄ as symmetries of the tetrahedron, a pyramid with triangular base and all edge lengths equal. Thus, there are 3 vertices around the base plus one at the apex, and any permutation of these 4 vertices corresponds to a symmetry of the tetrahedron.
      Recall that we can characterize a normal subgroup of G algebraically as a subgroup H which is a union of conjugacy classes of G (so that H goes to itself under conjugation).   Geometrically, a normal subgroup of G = Sym(X) is H = Sym(X^~), where X^~ is a decoration of X that is defined without a labelling of X (i.e., not referring to "vertex 1" or "the bottom", etc.).
      Problem: Find all the normal subgroups of S₄, algebraically by analyzing the conjugacy classes, and geometrically by decorating the tetrahedron.
3. Extra Credit: Show that any group of order 8 must be isomorphic to one of the groups listed in HW 10/29 #14.
4. Artin Ex 5.1.3, p.188 (cf. p.156). Write the symmetries of the given figures by putting them into the complex plane C, and writing symmetries as complex functions. For example, the symmetries of Fig 6.1.2, p. 156, are G = {z --> z + n for n ∈ Z} . (Assume that the spacing between each little runner is 1.)
5. Artin Ex 5.2.15, p.189 (cf. p.158).
6. Artin Ex 5.3.4, p.189 (cf. p.165).
7. Artin Ex 5.4.9, p.190.

1. Let V be a vector space over a field F. Show that if av = 0 for a ∈ F and v ∈ V, then a = 0 or v = 0 .
2. Prove that the span of a set of vectors is a subspace.
3. Show that a single vector {v₁} is linearly independent unless v₁ = 0 .Show that two vectors {v₁ , v₂} are linearly independent unless one is a scalar multiple of the other.
4. Show that {v₁ ,..., v_n} is a basis if and only if for each v ∈ V, there are unique scalars a₁ ,..., a_n ∈ F such that a₁v₁ + ... + a_nv_n = v .
5. Artin Ex. 3.3.1, p. 105. Given 4 spanning vectors for a 3-dimensional subspace W in R⁴ , find a basis for W . Hint: Span(w₁,w₂,...,w_n)=Span(w₁+aw_i,w₂,...,w_n)for any scalar a and any w_i . Reduce the given spanning set by such operations to get a simpler spanning set, for which it is clear which vector to drop from the spanning set. This is called Gaussian elimination.
6. Show that the space of n x n real matrices is a real vector space of dimension n² .
   Now do Ex. 3.3.5, p. 106. Notes:We write a matrix as M = (x_ij), where x_ij is the entry in row i, column j. The symmetric matrices are those with x_ij = x_ji for all i,j. Show that the symmetric matrices for a subspace of all matrices. Find a basis (and hence the dimension).
7. Ex. 3.3.9 and 3.3.10 about vector spaces over R versus C .
8. 3.3.11--13 about vector space over finite fields.
9. 3.3.16. Hint: Use the lemma from class: If {v₁,v₂,...,v_n} span a space W (say, a subspace of V), and {v₁,v₂,...,v_k} are linearly independent for some value k ≤ n , then we can remove some of the remaining elements v_k+1,...,v_n to get a basis for W.

1. For two bases A and B, we define the change of basis matrix to be the n × n matrix P_B^A such that:[v]_B = P_B^A * [v]_A , where * is matrix multiplication. Show that such a matrix is unique: that is, this matrix is determined by the bases A and B.Hint: the column vectors of P_B^A are P_B^A * (0,...,1,...,0)^t .
2. Show that:P_C^A = P_C^B * P_B^A   and  P_A^B = (P_B^A)^-1 .Hint: clearly P_A^A = I   the identity matrix.
3. Suppose we have a standard basis E and we are given the bases A, B as lists of vectors in the E-coordinates, namely as the matrices P_E^A andP_E^B . Then we can write the change of basis from A to B as:P_B^A = (P_E^B)^-1 * P_E^A .Now let V = R², and consider the bases:A = { (1,2)^t , (3,4)^t }   and   B = { (5,6)^t , (7,8)^t } Suppose v = (9,10)^t, and compute[v]_A and [v]_B using change-of-basis matrices.
4. Inside V = R³, consider the subspace W with basis A = { (1,2,3)^t , (4,5,6)^t } . Use column reduction to find a nice basis for W of the form:B = { (1,0,x)^t , (0,1,y)^t } .Let v = (5,4,3) ∈ W . Compute [v]_B directly by inspection, and use a change-of-basis matrix to compute [v]_A .
5. Ex. 3.4.9, p.107. The map B --> P_E^B gives a one-to-one map from the set B of all bases of V to the the group of all n × n invertible matrices GL_n(F) .Hint: Show that P_E^B is always an invertible matrix; and find an inverse mapping from invertible matrices to bases.
6. Ex. 3.4.11, p.107. Count the ordered bases of a finite vector space. What if we ignore the ordering?
   Extra Credit: Ex. 3.3.14: Count the distinct k-dimensional subspaces of an n-dimensional vector space over F_p.

1. Given a vector space V with basis B, and W with basis C, we define the matrix of a linear mapping T:V→W by:[T]_C^B * [v]_B= [T(v)]_Cwhere * means matrix multiplication. Show that there does indeed exist a unique matrix with this property, and its columns are given by the column vectors[T(b_i)]_C .
   Use the definition to show that if S:U→V and T:V→W , where U, V, W have respective bases B, C, D, then:[TS]_D^B = [T]_D^C * [S]_C^B .
2. Ex 4.1.1 & 4.1.2, p.145. Compute the kernel and image of a matrix T, or its rank. Hint: row reduction does not change the kernel of T ; column reduction does not change the image of T .
3. Ex 4.2.1, p.145.
4. Ex 4.2.2, p.145
5. Ex 4.2.4, 4.2.5, p.146
6. Ex 4.2.8, p.146: a matrix has the same rank as its transpose.Hint: Rank is a property of a linear mapping, so it does not change under change of basis. Reduce the matrix to standard form as in Prop. 2.9, p.114.
7. Ex. 4.2.9. The vector space of linear transformations = the vector space of matrices.
8. Extra Credit: The set of all k-dimensional subspaces of an n-dimensional vector space over F_p is called the Grassmannian Gr(k, F_pⁿ). We introduced the p-integer: [n]_p = (pⁿ - 1)/(p - 1)= pⁿ + p^n-1 + ... + p + 1 ,as well as the p-factorial and the p-binomial coefficient: [n]_p^! = [n]_p [n-1]_p ... [2]_p [1]_p   and   [n || k]_p =[n]_p^! /  [k]_p^![n-k]_p^!These should be thought of as functions of p, into which we can substitute particular values. Indeed, substituting p = 1 makes [n]_p = n ,   [n]_p^! = n! , and [n || k]_p =[n || k] , which is my notation for an ordinary binomial coefficient, the number of k-subsets of an n-set.
   1. Show that [k]_p^! is the number of ordered bases of a k-dimensional space, i.e., the number of invertible matrices #GL_p(F_p) ; that [n]_p^! / [n-k]_p^! is the number of lists of k independent vectors in F_pⁿ ; and [n || k]_p is the number of distinct k-dimensional subspaces of F_pⁿ .
   2. The p-binomial coefficients have many properties which parallel the ordinary binomial coefficients [n || k]. For example Pascal's triangle, which depends on the recurrence: [n || k] = [n-1 || k] + [n-1 || k-1]. Proof: A k-subset of {1,2,...,n} either does not contain n , in which case it is a k-subset of {1,2,...,n-1}; or removing n leaves a (k-1)-subset of {1,2,...,n-1}. A straight algebraic proof is also possible.
      Find and prove an appropriate analog of Pascal's triangle for [n || k]_p . Hint: you need to multiply by some p's to make it work.
   3. Prove that [n || k]_p is always a polynomial function of p , meaning that the denominator always divides the numerator as polynomials. Hint: Use the previous problem. Is there a more direct way?
   4. There is an amazing combinatorial formula which emphasizes the relationship between ordinary binomial coefficients (which count k-subsets) and p-binomial coefficients (which count k-subspaces).To get some idea of this formula, look at the example k = 1 , where there are n subsets S = {1}, {2}, ... , {n}, and the number of subspaces is a polynomial with n terms: [n || 1]_p = 1 + p + ... + p^n-1 .
      The general idea is that each k-subset S should contribute a term to [n || k]_p of the form p^r for some r.That is, we should be able to define a "weight" r = wt(S) so that:[n || k]_p   =   Σ_S p^wt(S) ,where the sum is over all k-subsets S.
      Define the right weight function, and prove your result.
9. Let G = GL_n(F) be the group of all invertible n × n matrices, with the operation of matrix multiplication. Prove that the center Z = Z(G) , the subgroup of elements which commute with all other elements of G , is just the set of non-zero scalar multiples of the identity matrix.
   Extra Credit: For a finite field F = F_p, the quotients GL_n(F) / Z make an important family of finite simple groups (having no normal subgroups). Can you prove this group is simple in any cases?