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PART C

Answer All Questions. Each Question carries 3 marks.
Show that inverse of an element a in the group 15 unique.
Show that (G,+6¢) is acyclic group where © = {0,1,2,3,4,5}
A= (2, 3, 4,6,12,18,24,36} with partial order of divisibility. Determine the POSET is
a lattice.
Consider the lattice 190 and [230 of all positive integer divisors of 20 and 30

respectively, under the partial order of divisibility. Show that is a Boolean algebra.

PART D
Answer any two Questions. Each Question carries 9 marks

a. Prove that the order of each subgroup of a finite group G is a divisor of the order of
the group G. (4.5)
b. Show that the set{ 0, 1, 2,3,4,5 }is a group under addition and multiplication
modulo 6. (4.5)

. a. Prove that every finite integral domain is a field. (4.5)

0. Show that (7, 0, ©) is a ring where 8 0 9 = at+b-1 andaO ४ = 2+0-व (4.5)

a. Consider the Boolean algebra D30.Determine the following:
i) All the Boolean sub-algebra of 1230.
ii) All Boolean algebras which are not Boolean sub-algebras of [230 having atleast four
elements. (4.5)

b. Consider the Lattice L in the figure. Find the L is distributive and complemented

lattice. Also find the complement of a,b,c. (4.5)
1
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6
8 b
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