Group Theory MCQs

1

Mathematics • Group Theory

Which one of the following is a group? Which one of the following is a group?

A. The set of all 2 by 2 matrices under multiplication. The set of all 2 by 2 matrices under multiplication. B. (R+, x), where a x b = sqrt(ab) for all a, b in R+. (R+, x), where a x b = sqrt(ab) for all a, b in R+. C. (C[0,1], x), where (f x g)(x) = f(x)g(x). (C[0,1], x), where (f x g)(x) = f(x)g(x). D. The set of odd integers under addition. The set of odd integers under addition.

सही उत्तर: C Correct Answer: C

Option C is the answer marked in the source paper. It represents the stated collection of continuous functions under pointwise multiplication.

2

Mathematics • Group Theory

Which of the following is true? Which of the following is true?

A. Every permutation can be written as a product of intersecting cycles. Every permutation can be written as a product of intersecting cycles. B. Every permutation can be written as a product of cycles of length 1. Every permutation can be written as a product of cycles of length 1. C. Every permutation can be written as a product of disjoint cycles. Every permutation can be written as a product of disjoint cycles. D. Every permutation can be written as a product of cycles that do not commute. Every permutation can be written as a product of cycles that do not commute.

सही उत्तर: C Correct Answer: C

Every permutation has a decomposition into disjoint cycles, unique up to the order in which those disjoint cycles are written.

3

Mathematics • Group Theory

Which one of the following groups is cyclic? Which one of the following groups is cyclic?

A. (Q, +) (Q, +) B. (Q+, multiplication) (Q+, multiplication) C. GL_n(R) under matrix multiplication GL_n(R) under matrix multiplication D. {[[1,n],[0,1]] : n belongs to Z} under matrix multiplication {[[1,n],[0,1]] : n belongs to Z} under matrix multiplication

सही उत्तर: D Correct Answer: D

The matrix [[1,n],[0,1]] is the nth power of [[1,1],[0,1]]. Thus the group is generated by one element and is cyclic.

4

Mathematics • Group Theory

Which of the following is a non-abelian group? Which of the following is a non-abelian group?

A. The set of non-zero real numbers under multiplication. The set of non-zero real numbers under multiplication. B. The real numbers under addition. The real numbers under addition. C. The set of 2 by 2 matrices under addition. The set of 2 by 2 matrices under addition. D. The set of 2 by 2 matrices with determinant 1 under matrix multiplication. The set of 2 by 2 matrices with determinant 1 under matrix multiplication.

सही उत्तर: D Correct Answer: D

The determinant-one matrices form SL(2), and matrix multiplication in this group is generally noncommutative.

5

Mathematics • Group Theory

What is the order of (2,6) in Z_6 x Z_12? What is the order of (2,6) in Z_6 x Z_12?

A. 1 1 B. 6 6 C. 10 10 D. 2 2

सही उत्तर: B Correct Answer: B

The order of 2 in Z_6 is 3 and the order of 6 in Z_12 is 2. The order of the pair is lcm(3,2)=6.

6

Mathematics • Group Theory

GL_n(R) denotes the group of n x n non-singular matrices under matrix multiplication. Which one of the following gives an abelian subgroup? GL_n(R) denotes the group of n x n non-singular matrices under matrix multiplication. Which one of the following gives an abelian subgroup?

A. Set of all non-singular upper-triangular matrices. Set of all non-singular upper-triangular matrices. B. Set of all non-singular matrices with positive trace. Set of all non-singular matrices with positive trace. C. Set of all matrices with positive determinant. Set of all matrices with positive determinant. D. Set of all non-singular diagonal matrices. Set of all non-singular diagonal matrices.

सही उत्तर: D Correct Answer: D

Non-singular diagonal matrices are closed under multiplication and inversion, and diagonal matrices commute with one another. Hence they form an abelian subgroup.

7

Mathematics • Group Theory

What is the order of an element g of a cyclic group G? What is the order of an element g of a cyclic group G?

A. The smallest positive integer r such that g^r is the identity. The smallest positive integer r such that g^r is the identity. B. The order axioms imposed on the group. The order axioms imposed on the group. C. The number of elements in the group. The number of elements in the group. D. The number of elements a in G such that a^2=e, the identity. The number of elements a in G such that a^2=e, the identity.

सही उत्तर: A Correct Answer: A

By definition, the order of g is the least positive integer r for which g^r=e, provided such an integer exists.

Group Theory MCQs

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