This is a 24-hour open book exam. Please attempt all questions. Q1, 2
and 3 are standard exam-type questions: Q4 is considerably harder but
only makes use of material we have seen in lectures or problem sheets.
1. (a) (i) Define what it means to say that a relation R on a set X is an
equivalence relation, explaining the terms you use.
(ii) Let X = M2(R) be the set of 2 × 2 matrices with real entries
and define R on X by ARB if B = PAQ for some invertible matrices
P,Q ∈ X. Prove that R is an equivalence relation.
(iii) Give a complete set of representatives for R on X and give an
explicit description of each equivalence class, with brief justification.
(b) Let S = {a+ b√−17 : a, b ∈ Z}.
(i) Define the norm N(s) of an element s ∈ S and show that N(st) =
N(s)N(t) for all s, t ∈ S.
(ii) An element u ∈ S is a unit if there exists v ∈ S such that uv = 1.
Prove that the only units are ±1.
(iii) An element s ∈ S is an S-prime if s = tv (t, v ∈ S) implies that one
of t,v is a unit. Prove that 2, 3, 1 +
√−17 and 1−√−17 are S-primes.
(iv) State the unique factorisation theorem for Z. By considering the
factorisations of 18 into S-primes in S, show that that the correspond-
ing theorem does not hold for S, i.e. S does not have unique factorisa-
tion.
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2. (a)(i) State (do not prove) Lagrange’s Theorem; deduce that in a finite
group the order of any element divides the order of the group.
(ii) Hence show that if p is a prime then in Z∗p, a
p−1 = 1 for any a ∈ Z∗p.
(iii) Solve x5 ≡ 3 (mod 19).
(b)(i) Let G and H be groups. Define what it means to say that a map
φ : G −→ H is a group isomorphism and what it means to say that G
and H are isomorphic.
(ii) Let G = C∞ and H = {
(
1 a
0 1
)
: a ∈ Z} under matrix multiplica-
tion. Prove that G and H are isomorphic.
3. (a) Let A = (aij) be an n × n matrix. Give the definition of det(A),
the determinant of A. State, without proof, the effect of each type of
elementary row operation on the determinant.
(b) (i) Evaluate the determinant of the matrix
A =

a b b b
b a b b
b b a b
b b b a

(ii) For which values of a and b is the matrix A invertible?
(c) Let A =

1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
.
Find an invertible matrix P such that P−1AP = D is diagonal, stating
what D is.
CONTINUED
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4. Lagrange’s Theorem says that if G is a finite group of order n, then
the order of any subgroup H of G divides n. However, it is not true
that if G is a group of order n and m is a number dividing n, then G
necessarily has a subgroup of order m and this question provides an
example of this.
(a) Let G be any group of order 6.
(i) Show that G cannot contain two distinct elements a,b of order 2
with ab = ba.
(ii) Show that it is not possible that all non-identity elements of G are
of order 2. (Hint: what happens if a, b and ab are all elements of order
two?)
(iii) Show that it is not possible that all non-identity elements of G are
of order 3. (Hint. If H and K are two subgroups of order 3, what do
you know about |H ∩ K|? What does this tell you about the size of
the set H ∪K?)
(iv) Hence show that either (a) G contains an element of order 6 or (b)
G contains two elements a, b of order 2 such that ab 6= ba.
(b) A4 is the group of even permutations of {1, 2, 3, 4}.
(i) Write down all elements of A4 and state their order.
(ii) Using (a)(iv), show that A4 does not contain a subgroup of order
6.
END OF PAPER
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