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Appendix A5

A Second Year Abstract AlgebraExamination Paper

ABSTRACT ALGEBRA

Duration one hour 30 minutes

Answer 3 out of 5 questions

Instructions to candidates1. Normal university regulations apply

2. The use of calculators is NOT permitted3. All questions carry equal marks

4. Show all working and explain your reasoning at all times

Materials provided1. Script answer book

2. Formulae sheet 2

1. For the setsA={1, 3, 7, 9}

B={1, 2, 4, 7, 8, 11, 13, 14}verify the equality

(A \B) (B\A) (A B) = A BProve this result for general sets, A, B. Show that A\B, B \A, and ABare disjoint sets and henceshow that

|A

B

|=

|A

|+

|B

| |A

B

|Describe briefly an application of this result.

(Total: 20 marks)

2. Explain what is meant by an equivalence relation and equivalence classes.

Form, nZ show that the relationRdefined by

mRn if mn(mod6) (i.e.:m nis divisible by 6)

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is an equivalence relation, and describe the corresponding equivalence classes.

Ifm and n belong to the same equivalence class ofR show that m + n is in the same equivalenceclass for allm,n and describe this equivalence class, illustrating the result with a particular choice

ofmandn.

(Total: 20 marks)

3. Define what is meant by an associative commutative binary operation, giving an example.

Prove that, defined on Q (The set of rational numbers) by

a b= a + b + 3ab

is a commutative binary operation. Is it associative?

Determine an identity element admitted by and show that it is unique. Show that with respect tothis identity the inversea1 ofais given by

a1 = a1 + 3a

Give an elementaQ that has no inverse with respect to .Show that the set of rational numbers is a not a group with respect to the binary operation . Finda subsetSofQthat is a group with respect to .

(Total: 20 marks)

4. Prove Lagranges Theorem - the order of every subgroupHof a finite group G of ordern is a divisorofn.

Show that the set

G={1,1, i,i} (i= 1)forms a group with respect to multiplication of complex numbers. Obtain a non-trivial subgroupH, justifying your choice, and use it to illustrate Lagranges Theorem.

(Total: 20 marks)

5. Describe, in about 150 - 200 words, the method of encryption and decryption in the RSA algo-rithm, defining any notation used.

If the message 11 is transmitted using modulus 33 and encryption integer e = 7, determine thecoded message sent and check your result by decoding.

Comment on the security of this message.

(Total: 20 marks)

END OF EXAMINATION PAPER