Practice Questions for CSIR NET Group Theory : Groups and Their Properties I

1. Let \(\sigma: \{1,2,3,4,5\} \rightarrow \{1,2,3,4,5\}\) be a permutation (one-to-one and onto function) such that \(\sigma^{-1}(j) \leq \sigma(j)\) for all \(j, 1 \leq j \leq 5\). Then which of the following are true?

(a.) \(\sigma \circ \sigma(j) = j\) for all \(j, 1 \leq j \leq 5\).
(b.) \(\sigma^{-1}(j) = \sigma(j)\) for all \(j, 1 \leq j \leq 5\).
(c.) The set \(\{k : \sigma(k) \neq k\}\) has an even number of elements.
(d.) The set \(\{k : \sigma(k) = k\}\) has an odd number of elements.

2. Which of the following is a non-cyclic group?

(a.) \((p\mathbb{Z}, +)\), where \(p\) is prime.
(b.) \((m\mathbb{Z}, +)\), where \(m\) is an integer.
(c.) \((\mathbb{R}^*, \cdot)\), where \(\mathbb{R}^*=\mathbb{R}-\{0\}\).
(d.) None of the above.

3. Let \(\mathbb{C}^*\) denote the multiplicative group of non-zero complex numbers. Let \(G_1\) be the cyclic subgroup generated by \(1+i\) and \(G_2\) be the cyclic subgroup generated by \(\frac{1+i}{\sqrt{2}}\). Which one of the following is correct?

(a.) Both \(G_1\) and \(G_2\) are infinite groups.
(b.) \(G_1\) is finite, but \(G_2\) is an infinite group.
(c.) \(G_2\) is finite, but \(G_1\) is an infinite group.
(d.) Both \(G_1\) and \(G_2\) are finite groups.

4. Let \(\sigma\) be an element of the permutation group \(S_5\). Then the maximum possible order of \(\sigma\) is

(a.) 5
(b.) 6
(c.) 10
(d.) 15

5. Let the group \(G=\mathbb{R}\) (the set of all real numbers) under addition and the group \(H=\mathbb{R}^{+}\) (the set of all positive real numbers) under multiplication.

(a.) \(H\) is a cyclic group and \(G\) is a non-cyclic group.
(b.) \(G\) is a cyclic group and \(H\) is a non-cyclic group.
(c.) Neither \(G\) nor \(H\) is cyclic.
(d.) Both \(G\) and \(H\) are cyclic.

6. Suppose that \(H\) is the smallest subgroup of \(\mathbb{Z}\) under addition, and \(H\) contains 18, 80, and 40. Then \(H\) is

(a.) \(18\mathbb{Z}\)
(b.) \(40\mathbb{Z}\)
(c.) \(2\mathbb{Z}\)
(d.) \(\mathbb{Z}\)

7. If the order of every non-identity element in a group is 'n', then

(a.) 'n' is necessarily a prime number.
(b.) 'n' can be any odd number.
(c.) 'n' is an even number.
(d.) 'n' can be any positive integer.

8. Let \(G=\{g_1, g_2, \ldots, g_n\}\) be a finite group, and suppose it is given that \(g_i^2 = \text{identity}\) for \(i=1,2,\ldots,n-1\). Then

(a.) \(g_n^2\) is the identity, and \(G\) is abelian.
(b.) \(g_n^2\) is the identity, but \(G\) could be non-abelian.
(c.) \(g_n^2\) may not be the identity.
(d.) None of the above.

9. Consider the multiplicative group \(G\) of all the (complex) \(2^n\) th roots of unity, where \(n=0,1,2,\ldots\). Then

(a.) Every proper subgroup of \(G\) is finite.
(b.) \(G\) has a finite set of generators.
(c.) \(G\) is cyclic.
(d.) Every finite subgroup of \(G\) is cyclic.

10. Which of the following numbers can be orders of permutations \(\sigma\) of 11 symbols such that \(\sigma\) does not fix any symbol?

(a.) 18
(b.) 30
(c.) 15
(d.) 28

11. Let \(G=\mathbf{Z}_{10} \times \mathbf{Z}_{15}\). Then

(a.) \(G\) contains exactly one element of order 2.
(b.) \(G\) contains exactly 5 elements of order 3.
(c.) \(G\) contains exactly 24 elements of order 5.
(d.) \(G\) contains exactly 24 elements of order 10.

12. Which of the following groups has a proper subgroup that is NOT cyclic?

(a.) \(\mathbf{Z}_{15} \times \mathbf{Z}_{17}\)
(b.) \(S_3\)
(c.) \((\mathbf{Z}, +)\)
(d.) \((\mathbb{Q}, +)\)

13. Let \(G_1\) be an abelian group of order 6 and \(G_2=S_3\). For \(j=1,2\), let \(P_j\) be the statement: " \(G\), has a unique subgroup of order." Then

(a.) Both \(R\) and \(P_2\) hold.
(b.) Neither \(R_1\) nor \(P_2\) hold.
(c.) \(P_1\) holds but not \(P_2\).
(d.) \(P_2\) holds but not \(P_1\).

14. Let \(D_8\) denote the group of symmetries of a square (dihedral group). The minimal number of generators for \(D_8\) is

(a.) 1.
(b.) 2.
(c.) 4.
(d.) 8.

15. Consider the following statements:

S: Every non-abelian group has a nontrivial abelian subgroup.

T: Every nontrivial abelian group has a cyclic subgroup. Then

(a.) Both \(S\) and \(T\) are false.
(b.) \(S\) is true and \(T\) is false.
(c.) \(T\) is true and \(S\) is false.
(d.) Both \(S\) and \(T\) are true.