Seminar assignments - Class exercises 1-6 PDF

Preview

Summary

Class Exercises 1-6...

Description

Groups and Rings III 2010 Class Exercise 1. Please hand up solutions in the lecture on Thursday 12th March.

1. Consider the group GL(2, R) =

("

a c

b d

#  )   a, b, c, d ∈ R 

of 2 × 2 invertible matrices with real entries with binary operation being matrix multiplication. For each of the following prove if they are, or are not, subgroups of G. If they are subgroups show if they are or are not abelian. (a) The set D of all diagonal matrices, i.e. b = c = 0 in the definition. (b) The set B of all upper triangular matrices, i.e. c = 0 in the definition. (c) The set H of all matrices whose determinant is π .

2. Consider the group U18 = {z ∈ C× | z18 = 1} with generator ω = exp(iπ /9). (a) What are the orders of ω9 and ω7 ? (b) Find all subgroups of U18 and draw the subgroup lattice.

3. Recall that in class we defined the quaternion group as H = {±1, ±i, ±j, ±k} with multiplication defined by letting the identity be 1, assuming that −1 commutes with everything else and that also ij = −ji = k, jk = −kj = i, ki = −ik = j, i2 = j 2 = k2 = −1 and ijk = −1. (a) Calculate the elements ik2 (1)ji and i3 j 2 (−1)k. (b) Find a subgroup of H of order 2. (c) Find a subgroup of H of order 4. (d) Find two elements of H that generate H. (Prove that they generate H).

Groups and Rings III 2010 Class Exercise 2. Please hand up solutions in the lecture on Thursday 26th March. 1. (a) If G is a finite group and g ∈ G show that g |G| = e. (b) Prove Fermat’s Little Theorem which says that if p is a prime and 0 < a < p then ap−1 ≡ 1 mod p . [Hint: Think about the group Zp×.] 2. Let g ∈ G and consider the function Adg : G → G defined by Adg (x) = gxg −1 . Show that Adg is an isomorphism for all g ∈ G. 3. Let x ∈ G. Prove that CG (x) = {x ∈ G | xg = gx } is a subgroup of G. 4. Determine the conjugacy classes of S4 . You can use the result from Lectures relating the conjugacy class of a permutation to its cycle structure. Pick an element π in each class and determine CS4 (π ). 5. (a) If G is a group show that Z(G), the centre of G, is a normal subgroup of G. (b) Find the centre of H. (c) Call a matrix in GL(n, C) a scalar matrix if it is a (non-zero) multiple of the identity matrix. Show that Z(GL(n, C)) is the group of scalar matrices. (Hint: Assume X is in the centre and consider the equation EX = X E where E is an elementary matrix as in Mathematics I. Try different kinds of elementary matrices. )

Groups and Rings III 2010 Class Exercise 3. Please hand up solutions in the lecture on Thursday 22nd April .

1. Find the commutator subgroup of the quaternion group H. Verify that it is normal. 2. Let H and K be groups and define a binary operation H ×K×H ×K → H ×K ((h1 , k1 )(h2 , k2 )) ֏ (h1 h2 , k1 k2 ) (a) Show that this binary operation makes H × K into a group. (b) Show that H0 = {(h, e) | h ∈ H} is a subgroup of H × K . (c) Show that the map ιH : H → H × K defined by ιH (h) = (h, e) is a one-to-one homomorphism with image H0 . (d) Show that the map πK : H × K → K defined by πK ((h, k)) = k is an onto homomorphism with kernel H0 .

3. If m and n are integers denote the least common multiple of m and n by lcm(m, n) and the greatest common divisor of m and n by gcd(m, n). Note that lcm (m, n) gcd(m, n) = mn. (a) If (h, k) ∈ H × K show that |(h, k)| = lcm(|h|, |k|). (b) If (m, n) ≠ 1 show that Cm × Cn 6≃ Cmn . (c) If (m, n) = 1 show that Cm × Cn ≃ Cmn .

4. Let U1 = {z ∈ C× | |z| = 1} < C× and R>0 = {x ∈ R× | x > 0}. Show that: (a) C× ≃ U1 × R>0 . (Hint: polar decomposition, i.e z = r exp(iθ)) (b) R× ≃ Z2 × R>0 .

Groups and Rings III 2010 Class Exercise 4. Please hand up solutions in the lecture on Thursday 13th May. 1. Consider the group U (1) = {z ∈ C× | |z| = 1}. Identify the orbits √ of the action of U (1) on C defined by (u, w) ֏ uw for u ∈ U (1) and w ∈ C. Find SU (1) (0) and SU(1) (1 + i 2). 2. Let G act on X . (a) Show that SG (x) is a subgroup of G. (b) Show that SG (gx) = gSG (x)g −1 . 3. Let a finite group G act on a finite set X. We say the action is free if for every x ∈ X the only g satisfying gx = x is g = e. (a) Show that if the action is free then SG (x) = {e} for all x ∈ X . (b) Show that if the action is free then Xg = ∅ for all g ≠ e. (c) Use Burnside’s theorem to show that if G acts freely on X then the number of orbits is |X |/|G|. 4. Let H < G be finite groups. (a) Define a function H × G → G by (h, g) ֏ h ⋆ g = gh−1 . Show that this is a free action of H on G. (b) Show that the orbits of this action of H on G are the left cosets of H . (c) Use Burnside’s theorem or the previous question to deduce Lagrange’s theorem that the number of left cosets of H in G is |G|/|H |. 5. We wish to paint each edge of a triangle with one of n different coloured paints. We are allowed to paint adjacent edges with the same coloured paint. Two paintings are considered to be the same if we can act by a symmetry of the triangle (i.e an element of S3 ) until they look the same. Use Burnside’s Theorem to show that the number of different paintings is n3 + 3n2 + 2n . 6 Exam 2009

Groups and Rings III 2010 Class Exercise 5. Please hand up solutions in the lecture on Thursday 27th May.

1. If R is a ring with identity show that the set of all units in R is a group under multiplication. 2. Consider the ring of real quaternions: R(H) = {x1 + x2 i + x3 j + x4 k | x1 , x2 , x3 , x4 ∈ R} We define the addition and multiplication by assuming everything is linear over the real numbers and using the usual rules of multiplications in the quaternion group. E.g. (5 + 2j)(i + 3k) = 5i + 15k + 2ji + 2jk = (5 + 2)i + (15 − 2)k = 7i + 13k and (1 + 3j) + (7i + 2j + k) = 1 + 7i + 5j + k. (a) If x = x1 + x2 i + x3 j + x4 k define x¯ = x1 − x2 i − x3 j − x4 k and show that xx ¯ = kx k2 where kx k is the usual Euclidean length of a vector (x1 , x2 , x3 , x4 ) ∈ R4 . (b) Deduce that any non-zero x ∈ R(H) is a unit. (c) Deduce that R(H) is a skew-field. √ √ 3. Consider the set Q( 7) = {a + b 7 | a, b ∈ Q} ⊂ Q. √ (a) Show that Q( 7) is a subring of Q. √ (b) Show that Q( 7) is a field.

4. Complete the following table. Ring

Commutative

Identity

Z Z(i) Z4 Z3 √ Q( 7) R(H) M2 (C)

yes

1

Units ±1

Zero Divisors none

Field no

Integral Domain yes

Note: Z(i) a, b ∈ Z} is the ring of Gaussian Integers, a subring of C. √ = {a + bi | √ Q( 7) = {a + b 7 | a, b ∈ Q} is a subring of R. R(H) see Question 2. You don’t have to prove everything. Just fill out the table. 5. Recall the construction in lectures of the field of quotients of an integral domain D which involved the set S = {(a, b) | a, b ∈ D, b ≠ 0}. (a) Show that the relation (a, b) ≃ (c, d) if ad = bc is an equivalence relation on S . (b) Show that the addition [(a, b)] + [(α, β)] = [(aβ + bα, bβ)] is well-defined.

Groups and Rings III 2010 Class Exercise 6. Not for handing in. 1. Consider the ring of Gaussian Integers, Z(i) = {a + bi | a, b ∈ Z} with Euclidean valuation δ(a + bi) = a2 + b 2 . (a) For a = 1 + 2i, b = 3 − i , find q, r ∈ Z(i) such that a = bq + r , with δ(r ) < δ(b), where δ is the Euclidean norm for Z(i). (b) For each of 2 and 3 either show that they are irreducible or factorise them into products of irreducibles in Z(i). 2. Recall that that the √ set of all units G in a ring with identity is a group with operation the ring multiplication. For the ring Z( 2) use this to show that the group of units is infinite. Hint: Find a unit using the fact that it has norm 1 and then show that it has infinite order in G. √ √ 3. Consider the integral domain D = {a + b −5 | a, b ∈ Z} with norm N(a + b −5) = a2 + 5b 2 . You may assume that N(αβ) = N (α)N(β) for all α, β ∈ D . (a) Prove that α ∈ D is a unit if and only if N (α) = 1. (b) Find all units of D . (c) Show that if N (α) = 9, then α is irreducible. √ √ (d) By considering the product (2 + −5)(2 − −5), show that 3 is not prime in D . (e) Is D a unique factorization domain? Justify your answer. (Hint: In case we haven’t got to it by the time you do this in a UFD primes are the same things as irreducibles.) (Exam 2008) √ √ √ 4. Consider the integral domain Z( 10) = {a + b 10 | a, b ∈ Z} with the norm N(a + b 10) = a2 − 10b 2 . √ (a) Use the norm to describe the units of Z( 10). √ (b) By considering a2 mod 10 show that for any x ∈ Z( 10) we have that N(x) mod 10 can only be 0, 1, 4, 5, 6, 9. √ √ √ (c) Using (b) show that 2, 3, 4 + 10 and 4 − 10 are irreducible in Z( 10). √ (d) Is Z( 10) a unique factorization domain ?...