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Equivalence Relation - Groups and Rings - Exam, Exams of Statistics

University of AllahabadStatistics

This is the Exam of Groups and Rings which includes Six Elements, Incomplete Cayley, Identity, Multiplicative Group, Real Entries, Subgroup, Cycle Notation, Products Of Transpositions, Alternating Group etc. Key important poiints are: Equivalence Relation, Binary Relation, Relation, Statements, Elements, Non Abelian, Isomorphic, Group Homomorphism, Map, Subgroup

Typology: Exams

2012/2013

Uploaded on 02/26/2013

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LANCASTER UNIVERSITY
2012 EXAMINATIONS
PART II (Second Year)
MAT HEMATICS & STATISTICS 1 hour
Math 226: Groups
You should answer ALL Section A questions and ONE Section B question.
SECTION A
A1. (a) Let Rbe a binary relation on a set X. Define what it means for Rto be an
equivalence relation on X.[3]
(b) Let X=R2. Define a relation Ron Xby: (x, y)R(u, v)⇔x−u=y−v. Show
that Ris an equivalence relation. [4]
A2. Let Gbe a group. Prove the following statements about elements x, y, z, a ∈G:
(a) (xy)−1=y−1x−1,[2]
(b) (xyx−1)=xy−1x−1[2]
(c) If xyx−1=z−1az then byb−1=awhere b=zx.[2]
A3. (a) Define the order o(g)ofanelementgof a finite group G.[1]
(b) Write down the order of each element of S3.[2]
(c) Solve the equation (1 3 2)g(1 2) = (1 3). [3]
(d) Show that S3is non-abelian. [2]
(e) Name a group of order 6 which is not isomorphic to S3.[1]
A4. (a) Define what is meant by a group homomorphism.[2]
(b) Show that the map φ:R→C∗,φ(x)=e2πix is a group homomorphism. [2]
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LANCASTER UNIVERSITY

2012 EXAMINATIONS

PART II (Second Year)

MATHEMATICS & STATISTICS 1 hour

Math 226: Groups

You should answer ALL Section A questions and ONE Section B question.

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. Prove the following statements about elements x, y, z, a ∈ G: (a) (xy)−^1 = y−^1 x−^1 , [2] (b) (xyx−^1 ) = xy−^1 x−^1 [2] (c) If xyx−^1 = z−^1 az then byb−^1 = a where b = zx. [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) Name a group of order 6 which is not isomorphic to S 3. [1]

A4. (^) (a) Define what is meant by a group homomorphism. [2] (b) Show that the map φ : R → C∗, φ(x) = e^2 πix^ is a group homomorphism. [2]

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SECTION A continued

A5. Decide whether each of the following questions 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.

SECTION B

B1. (^) (a) Let G = D 12 with the standard generators a, b satisfying a^6 = b^2 = e and bab−^1 = a−^1. Let C 2 = {± 1 }, a group under multiplication. Define a map ϕ : G → C 2 by: ϕ(aj^ ) = ϕ(baj^ ) = (−1)j^ for 0 ≤ j ≤ 5. (i) Show that ϕ is a surjective group homomorphism. [3] (ii) Determine the kernel K of ϕ. [1] iii) Show that G/K ∼= C 2. [1] (b) Let H be a subgroup of the group G. (i) Show that the subset ZG(H) = {g ∈ G : gx = xg ∀ x ∈ H} is a subgroup of G. [3] (ii) For a, b ∈ G show that axa−^1 = bxb−^1 for all x ∈ H if and only if b−^1 a ∈ ZG(H). [2] (iii) If H is abelian, show that H ≤ ZG(H). [1] (iv) Suppose in addition ZG(H) is cyclic. Is H cyclic? Briefly justify your answer. [1] (c) Let σ = (1 3 7 2 8 6)(4 3 1 5)(3 6 2 7), τ = (1 3 5 7 2 4 6 8)(1 2 3 4 5 6 7 8) ∈ S 8. (i) Determine the order of σ and the order of τ στ −^1. [3] (ii) Show that σ and τ are of different signs. [2] (iii) Define the alternating group A 8. Show that σ^4 τ 2 σ^2 ∈ A 8. [2]

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