MTH212 Abstract Algebra B.Sc Question Bank : nmu.ac.in

Name of the University : North Maharashtra University
Degree : B.Sc
Department : Mathematics
Year : II
Name Of The Subject : MTH212 Abstract Algebra
Document type : Question Bank
Website : nmu.ac.in

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NMU Abstract Algebra Model Paper

NORTH MAHARASHTRA UNIVERSITY,
Question Bank New syllabus w.e.f. June 2008
Class : S.Y. B. Sc.
Subject : Mathematics
Paper : MTH – 212 (A) Abstract Algebra

Related : North Maharashtra University Calculus of Several Variables B.Sc Question Bank : www.pdfquestion.in/5275.html

Unit – I

2 Marks

1) Define product of two permutations on n symbols. Explain it by an example on 5 symbols.
2) Define i) a permutation ii) a symmetric group.
3) Define i) a cycle ii) a transposition.
4) Let C1 = (2 3 7) , C2 = (1 4 3 2) be cycles in S8. Find C1C2 and express it as product of transpositions.

5) For any transposition (a b) ? Sn , prove that (a b) = (a b)-1 .
6) Prove that every cycle can be written as product of transpositions.
7) Define disjoint cycles. Are (1 4 7) , (4 3 2) disjoint cycles in S8?

8) Write down all permutations on 3 symbols {1, 2, 3}.
9) Prove that An is a subgroup of Sn.
10) Let f be a fixed odd permutation in Sn (n > 1). Show that every odd permutation in Sn is a product of f and some permutation in Sn.

1 Marks

Choose the correct option from the given options.
1) Let A , B be non empty sets and f : A = B be a permutation . Then – – –
a) f is bijective and A = B
b) f is one one and A = B
c) f is bijective and A = B
d) f is onto and A = B

2) Let A be a non empty set and f : A -> A be a permutation . Then – – –
a) f is one one but not onto
b) f is one one and onto
c) f is onto but not one one
d) f is neither one one nor onto

3) Cycles (2 4 7) and (4 3 1) are – – –
a) inverses of each other
b) disjoint
c) not disjoint
d) transpositions

4) Every permutation in An can be written as product of – – –
a) p transpositions, where p is an odd prime
b) odd number of transpositions
c) even number of transpositions
d) none of these

5) The number of elements in Sn = – – –
a) n
b) n!
c) n!/2
d) 2n

6) The number of elements in A6 = – – –
a) 6
b) 720
c) 360
d) 26

4 Marks

1) Let g = SA , A = {a1 , a2 , – – – , an}. Prove that
i) g-1 exists in SA.
ii) g g-1 = I = g-1 g , where I is the identity permutation in SA.

2) Let A be a non empty set with n elements. Prove that SA is a group with respect to multiplication of permutations.

3) Let Sn be a group of permutations on n symbols {a1 , a2 , – – – , an}. prove that o(Sn) = n!. Also prove that Sn is not abelian if n = 3.
4) Define a cycle. Let a = (a1 , a2 , – – – , ar-1 , ar) be a cycle of length r in Sn. Prove that a -1 = (ar , ar-1 , – – – , a2 , a1).

5) Prove that every permutation in Sn can be written as a product of transpositions.
6) Prove that every permutation in Sn can be written as a product of disjoint cycles.

7) Define i) a cycle ii) a transposition. Prove that every cycle can be written as a product of transpositions.
8) Let f , g be disjoint cycles in SA. Prove that f g = g f.

9)Prove that there are exactly n!/2 even permutations and exactly n!/2 odd permutations in Sn (n>1).
10)Prove that for every subgroup H of Sn either all permutations in H are even or exactly half of them are even.

11) If f , g are even permutations in Sn then prove that f g and g-1 are even permutations in Sn
12) Define an odd permutation. Let H be a subgroup of Sn, (n>1), and H contains an odd permutation. Show that o(H) is even.

23) Compute a-1ba where a = (2 3 5)(1 4 7), b = (3 4 6 2) ? S7. Also express a-1ba as a product of disjoint cycles.
24) Show that there can not exist a permutation a = S8 such that a(1 5 7)a-1 = (1 5)(2 4 6).
25) Show that there can not exist a permutation a = S9 such that a(2 5)a-1 = (2 7 8).
26) Show that there can not exist a permutation µ = S8 such that µ(1 2 6)(3 2)µ-1 = (5 6 8).
27) Show that there can not exist a permutation a = S7 such that a-1(1 5)(2 4 6)a = (1 5 7).