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LANCASTER UNIVERSITY
2012 EXAMINATIONS
PART II (Second Year)
MATHEMATICS & STATISTICS 2 hour
Math 225: Groups and Rings
You should answer ALL Section A questions and THREE Section B questions. In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40.
SECTION A
A1. (^) (a) Let R be a binary relation on a set X. Define what it means for R to be an equivalence relation on X. [3] (b) Let X = R^2. Define a relation R on X by: (x, y)R(u, v) ⇔ x − u = y − v. Show that R is an equivalence relation. [4]
A2. Let G be a group.
(a) Prove that (xy)−^1 = y−^1 x−^1 for any x, y ∈ G. [2] (b) Hence or otherwise deduce that (xyx−^1 )−^1 = xy−^1 x−^1 , [2]
A3. (^) (a) Define the order o(g) of an element g of a finite group G. [1] (b) Write down the order of each element of S 3. [2] (c) Solve the equation (1 3 2)g(1 2) = (1 3). [3] (d) Show that S 3 is non-abelian. [2] (e) Give an example of a group of order 6 which is not isomorphic to S 3. [1]
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SECTION A continued
A4. Decide whether each of the following statements is true or false. Briefly justify your answer. [4]
(a) If H is a subgroup of a group G then the right cosets of H are the same as the left cosets of H. (b) For any group G, the subset {g ∈ G : g^2 = e} is a subgroup of G.
A5. For each of the following statements, decide whether it is true or false. Briefly justify your claim.
(a) Every subring of a ring R is an ideal in R. (b) 14 is the multiplicative inverse of̂ 11 in̂ Z 51.
(c) The set
a 0 0 b
: a, b ∈ R
is a unital ring. [6]
A6. Let R be a commutative ring and a, b ∈ R. (a) Prove that the set Ra = {ra : r ∈ R} is an ideal in R. (b) Define the expression “b divides a”. (c) Suppose that R is unital. Prove that if b divides a, then Ra ⊆ Rb. [7]
A7. Consider the ring R = M 2 (C) of 2 × 2 matrices with coefficients in C, and let
S =
a 0 b c
: a, b, c ∈ C
and I =
a 0
: a ∈ C
(a) Prove that S is a unital subring of R. Hence, give the multiplicative identity element of S. (b) Prove that I is an ideal in S. (c) Decide whether I is a prime ideal in S and prove your claim. [13]
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SECTION B continued
B2. (^) (a) (i) Define an automorphism of a group G. [2] (ii) Show that if θ and ϕ are automorphisms of G then so are θ ◦ ϕ and θ−^1. [3] (iii) Hence show that the set Aut G of automorphisms of G forms a group under composition. [4] (b) Let G be the group of invertible 2 × 2 matrices with real coefficients (with the operation of multiplication) and let
X =
a b −b a
: a, b ∈ R, a^2 + b^2 = 1
(i) Show that X is a subgroup of G. [5]
Define a map ϕ : R → G by: ϕ(θ) =
cos θ sin θ − sin θ cos θ
(ii) Show that ϕ is a group homomorphism from the additive group of R to G. [3] (You may use the sine and cosine identities: cos(α + β) = cos α cos β − sin α sin β, sin(α + β) = sin α cos β + cos α sin β.) (iii) Determine the kernel and the image of ϕ. [2] (iv) Hence show that X ∼= R/ 2 πZ. [1]
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SECTION B continued
B3. Let R, S be rings and ϕ : R −→ S a ring homomorphism. (a) (i) Prove that ker(ϕ) is an ideal in R. (ii) Suppose that R is unital, with multiplicative identity 1. Prove that 1 ∈ ker(ϕ) if and only if ϕ = 0. [5] (b) Let I be an ideal in S and let ϕ−^1 (I) = {a ∈ R : ϕ(a) ∈ I} ⊆ R. (i) Prove that ϕ−^1 (I) is an ideal in R. (ii) Let ϕ : Z[X] −→ Z be the ring homomorphism defined by ϕ(f (X)) = f (0) (that is, ϕ is the evaluation at 0), and let I = 2Z ⊂ Z. Find ϕ−^1 (I). [6] (c) Let ϕ : C −→ M 2 (R) be the map defined by
ϕ(a + bi) =
a −b b a
for all a + bi ∈ C, with a, b ∈ R.
(i) Prove that ϕ is a ring homomorphism, and that ϕ is injective. (ii) Hence show that M 2 (R) has a subring isomorphic to C. [9]
B4. (a) Let R be a commutative unital ring, with multiplicative identity 1 and I an ideal in R. Prove that if 1 ∈ I, then I = R. [3] (b) Find the multiplicative inverse of 15 in̂ Z 53. [3] (c) Consider the polynomial ring R = Q[X]. Let f (X) ∈ R be a non constant polynomial, and write I = f (X)R for the ideal generated by f (X). (i) Prove that if I is a prime ideal in R then f (X) satisfies the following property: if g(X), h(X) ∈ R are such that f (X) = g(X)h(X), then either f (X) divides g(X), or f (X) divides h(X). (ii) Write R = R/I for the quotient ring. Prove that every coset ĝ(X) ∈ R with ĝ(X) = ̂0 contains a polynomial of degree less than the degree of f (X). (iii) Let f (X) = X^3 − X + 1 ∈ R and g(X) = X^5. Find r(X) ∈ R such that r̂(X) = ĝ(X) in the quotient ring R/f (X)R and such that deg(r(X)) ≤ 2. [14]
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