MATH202201 Groups and Vector Spaces 202021
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MATH202201
Groups and Vector Spaces
Semester One 202021
1. (a) Using the groups axioms decide whether each of the following sets and operations forms a group. Give reasons for your answers.
i. Q+ = {x e Q I x > 0} under multiplication.
ii. Q/{0} with the operation x * y = x/y for all x, y e Q/{0}.
iii. The pair (S, *) given by
S = { 
│m, n e Z}
where the operation * is multiplication.
(b) The elements in the following are in the symmetric group S6 .
i. Write the following 2-row notation element in cycle notation:
|
|
2 4 |
3 6 |
4 2 |
5 5 |
1(6)、. |
ii. Find the inverse of g = (165)(34) e S6 .
iii. Simplify the product of cycles:
(136)(542)(35)(46)(21).
(c) i. Show that there are 720 7-cycles in the symmetric group S7 .
ii. Prove that each 7-cycle is an even permutation.
iii. Let g be a 7-cycle in S7 . Show that the centraliser of g is cyclic.
iv. Find the number of elements of the conjugacy class of the 7-cycle g .
2. The following is a multiplication table for a group G. The product a * b is given in the table where a is from the leftmost column and b from the top row.
e v w x y z
e 
e v w x y z
v 
v w e z x
w 
w e v z x y
x 
x y z e w
y 
y x v w e
z 
z y w e v
(a) What are the products v * x, y * v , z * v, and x * y? (These are missing from the table.)
(b) What is the order of the group?
(c) Is the group Abelian? Give a reason for your answer.
(d) What are the inverses of z and x?
(e) What are the orders of the elements e, x, v, and y?
(f) Let H = {e, x}. Show that this is a subgroup of G.
(g) Calculate the right cosets of H .
(h) Without calculating the left cosets of H, explain why H is a normal subgroup of G.
(i) Construct a multiplication table for the quotient group G/H .
(j) Is G/H cyclic? Give reasons.
(k) Describe the subgroup Z < G/H generated by the element Hy e G/H .
(l) What is the order of (G/H) /Z?
3. (a) Show that Z55 × Z39 is isomorphic to Z2145 and that Z65 × Z91 is not isomorphic to Z5915 .
(b) i. Prove that if G is a group of order pq where p and q are prime numbers
with p 
q and H is a normal subgroup of G but is not the trivial subgroup generated by the identity element of G, then G/H is Abelian.
ii. Consider the statement:
‘If G is a group of order pq where p and q are prime numbers with p 
q and H is a normal subgroup of G, then G/H is Abelian. ’
Is this statement true? If so, then give a proof. If not, then provide a coun- terexample.
(c) Let R+ denote the group given by the set {x e R I x > 0} under multiplication of real numbers. Let T be the subgroup of C↓ = C/{0} given by T = {z e C I IzI = 1} under complex multiplication.
The map θ : C↓ → R+ × T is defined by
θ(z) = ╱ IzI, 
、
and the homomorphisms pr1 : R+ × T → R+ and pr2 : R+ × T → T are given by
pr1 (r, w) = r for all r e R+ and w e T
pr2 (r, w) = w for all r e R+ and w e T.
i. Show that θ is an isomorphism.
ii. Find the kernel and image for each of pr1 · θ and pr2 · θ .
iii. Sketch the cosets of C↓ / ker(pr1 · θ) and C↓ / ker(pr2 · θ).
4. (a) Let v扌 = (1, 2, -1, 1)T , v2 = (2, 0, 1, 1)T , v3 = (1, -6, 5, 0)T . Find a vector v4 such that {v扌 , v2 , v3 , v4 } is a basis of R4 .
(b) Let F be a field, V a vector space over F , and v扌 , . . . vn e V . Let φ : Fn → V be the map defined by φ((a1 , . . . , an )T ) = a1v扌 + . . . + anvn . Show that φ is an isomorphism of vector spaces if and only if {v扌 , . . . , vn } is a basis of V . (You may assume that φ is a linear transformation.)
(c) Let V be a vector space over R and T : V → V be a linear transformation which is injective but not surjective. Show that dim(V) is infinite, and give an example of such V and T . Also give an example of a linear transformation S : V → V which is surjective but not injective. State carefully any theorems which you quote.
(d) Show that for every n > 1 there is a linearly dependent subset of Rn of size n+1 all of whose proper non-empty subsets are linearly independent. Justify your answer in a few sentences.
(e) Recall that Pn (R) is the vector space over R of all polynomials a0 +a1 x+ . . . +an xn where a0 , . . . , an e R. Find the dimension m of the quotient space P4 (R)/P2 (R) and write down an isomorphism φ : P4 (R)/P2 (R) → Rm (you need not prove that your map is an isomorphism).
5. (a) Recall that Pn (R) is the vector space over R of all polynomials a0 +a1 x+ . . . +an xn where a0 , . . . , an e R. Define T : P2 (R) → P2 (R) by T (p(x)) = xp\ (x), where p\ (x) is the derivative of x. Show that T is a linear transformation, and find the matrix of T with respect to the basis x, 1 + x, x2 . Find also the matrix of the composition T · T with respect to this basis.
(b) Give an example of a 3 × 3 lower-triangular matrix A (i.e. with zeros above the main diagonal) over R which is not diagonalisable, explaining why it is not diagonalisable. You may apply a standard criterion for diagonalisability, but should indicate clearly how it is being used. Give the algebraic and geometric multiplicities of the eigenvalues of A.
(c) Let A = 
0(1) 5(0) 0(2)．(、) . Write down an orthogonal matrix P and a diagonal matrix
D such that P一1 AP = D . Show carefully your calculations. Using the diagonali- sation, find A20 (you should leave large powers of numbers in your answer, rather than calculating them).
(d) Using the determinant map, or otherwise, show that the special orthogonal group SO4 (R) is a normal subgroup of index 2 in O4 (R).
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2022-08-23