Analysis & Group theory problems
Solutions
1. Find the minimum value of f (x) = (2x2 5x + 2)3 over the closed interval [0, 1].
Since f is continuous on a closed interval, it suces to check the endpoints, the points
at which f does not exist and the point
Mathematics 1214: Introduction to Group Theory
Homework exercise sheet 7
Due 12:50pm, Friday 19th March 2010
1. Which of the following sets of real numbers contains a least element?
[Recall that if S R, then S contains a least element if there is an eleme
Mathematics 1214: Introduction to Group Theory
Solutions to homework exercise sheet 8
1. Let G be a group and let a, b G.
(a) Prove that if a, b G, then a = b ab1 = e.
(b) Prove that G is an abelian group if and only if aba1 b1 = e for all a, b G.
Solutio
Mathematics 1214: Introduction to Group Theory
Solutions to homework exercise sheet 10
1. (a) Determine, with justication (and without using the Fundamental Theorem of Abelian
Groups), which of the following groups are isomorphic, and which are not isomor
Mathematics 1214: Introduction to Group Theory
Solutions to homework exercise sheet 9
1. Let G and H be two groups. Prove that G H is abelian if and only if G is abelian and H is
abelian.
[Hint: for = , rst think about elements of the form (g, eH ).]
Solu
Mathematics 1214: Introduction to Group Theory
Solutions to exercise sheet 11
1. Let n 3 and let Dn be the dihedral group of order 2n. Writing = 2/n , show that is a
normal subgroup of Dn . [Hint: if r is reection in the x-axis, then r = 1 r, and the reec
Mathematics 1214: Introduction to Group Theory
Tutorial exercise sheet 4
1. The complex numbers C form a group under addition with identity element 0, such that the
inverse of z C is z.
Consider the ve sets N, Z, Q, R, C. Which of these are subgroups of (
Mathematics 1214: Introduction to Group Theory
Tutorial exercise sheet 3
1
1 2 3 4 5 6 7
, writing your answer in 2-row notation
3 1 7 6 5 4 2
and as a product of disjoint cycles. Is this an even or an odd permutation?
1. (a) Compute
(b) Let n N, and cons
Mathematics 1214: Introduction to Group Theory
Tutorial exercise sheet 2
1. Let be an operation on a set S. Suppose that e is an identity element for , and
that there are elements x, y1 , y2 S such that
x y1 = e = x y2 ,
y1 x = e = y2 x and y1 = y2 .
(a)
Mathematics 1214: Introduction to Group Theory
Tutorial exercise sheet 6
1. Let S be a non-empty set, and let EquivRels(S) be the set of all equivalence relations on S,
and let Partitions(S) be the set of all partitions of S.
(a) Given an equivalence rela
Mathematics 1214: Introduction to Group Theory
Solutions to tutorial exercise sheet 1
1. Let m, n N, and let S = cfw_1, 2, . . . , m and T = cfw_1, 2, . . . , n. How many
mappings are there from S to T ? How many of these are injective, and how many
are b
Mathematics 1214: Introduction to Group Theory
Tutorial exercise sheet 5
1. Let S and T be subsets of the plane P = R2 . Give examples to show that both of the statements
below are false.
(a) S T = M(S) M(T )
(b) S T = M(T ) M(S)
Solution (a) For example,
Mathematics 1214: Introduction to Group Theory
Tutorial exercise sheet 7
1. Let n N, and let : Z Z be the mapping (m) = m + 1 for m Z.
As usual, we write k = if k N, 0 = Z and we also dene k = (1 )k if k Z
with k < 0.
n times
(a) Show that k (m) = m + k f
Mathematics 1214: Introduction to Group Theory
Solutions to tutorial exercise sheet 9
1. Let G1 and G2 be two groups.
Prove that if H1 G1 and H2 G2 , then H1 H2 G1 G2 .
[Here, H1 H2 is the set cfw_(h1 , h2 ) : h1 H1 , h2 H2 .]
Solution If x H1 H2 then x =
Mathematics 1214: Introduction to Group Theory
Solutions to tutorial exercise sheet 8
1. For each element of the dihedral group D3 , compute the order o() and nd .
Solution Let be the identity mapping in D3 ; then o() = 1 (since it is the identity element
Mathematics 1214: Introduction to Group Theory
Solutions to tutorial exercise sheet 10
1. Which of the following groups is cyclic? As usual, you should prove that your answers are
correct.
(a) Z2 Z2
(b) Z3 Z2
(c) Z5 Z10
(d) Z Z
Solution (a) Every element
Mathematics 1214: Introduction to Group Theory
Homework exercise sheet 4
Due 12:50pm, Friday 19th February 2010
1. (a) Show that H = cfw_(1), (1 3 4), (1 4 3) is a subgroup of (S4 , ).
[Here, (1) represents the identity permutation in S4 ].
(b) Show that
Mathematics 1214: Introduction to Group Theory
Homework exercise sheet 6
Due 12:50pm, Friday 12th March 2010
1. Find M(T ) for the following sets T P . If M(T ) has nite order, give its Cayley table.
(a) The circle of radius 1 centred at the origin (this
Mathematics 1214: Introduction to Group Theory
Homework exercise sheet 1
Due 12:50pm, Friday 29th January 2010
1. List all of the mappings : cfw_1, 2 cfw_a, b. Which of these are injective, which
are surjective and which are bijective?
Solution These are:
MA1214 exam questions and solutions, Trinity term 2010
1. Let be an operation on a set G. Suppose that (G, ) is a group, and consider a
second operation on G, dened by
x
y = x y 1
for x, y G
where, as usual, y 1 denotes the inverse of y with respect to th
Problem session solutions
1. For the function f (x) = exp(x2 /2), dened for x R,
(a) On which intervals is f non-negative? On which is it non-positive?
(b) On which intervals is f increasing? On which is it decreasing?
(c) On which intervals is f convex?
Problem session solutions
1. Let P = R2 = cfw_
x
y
: x, y R denote the plane, and consider the relation on P dened by
p q p and q are linearly dependent.
(a) Show that is not an equivalence relation on P .
(b) Show that is an equivalence relation on P \ c
Selected problem session solutions
1. A function f (A, B), where A and B are n n-matrices, is dened by the formula
(a) f (A, B) = tr(AB);
(b) f (A, B) = tr(AB T );
(c) f (A, B) = tr(AB) + tr(A) tr(B).
For each of these functions, nd out which of the follo
Problem session solutions
1. The sequence an = (1)n is not convergent, but some of its subsequences are. Which ones?
Solution The ones with only nitely many odd n converge to 1. The ones with nitely many
even n converge to 1. None of the others converge.
Problem session solutions
1. Let G be a cyclic group, with identity element e. Then G = a for some a G. Let H be a
subgroup of G.
Prove:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
The set S = cfw_k N : ak H is either empty, or contains a least element.
If S = then H
Problem session solutions
1. Let G, H and K be three groups. Show that if : G H and : H K are two homomorphisms, then : G K is a homomorphism.
Solution Let = . Then is a well-dened mapping G K and for any a, b G, we
have
(ab) = (ab) = (a)(b) = (a)(b) = (a
Problem session solutions
1. Let a = (1 2 3 4) S4 and consider the subgroup H = a . Determine the right cosets of H
in S4 .
Solution We have H = cfw_(1), a, a2 , a3 = cfw_(1), (1 2 3 4), (1 3)(2 4), (1 4 3 2), and |S4 | = 24.
The right cosets partition S