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

16. In the group \((\mathbb{Z}, +)\), the subgroup generated by 2 and 7 is

(a.) \(\mathbf{Z}\).
(b.) 52.
(c.) \(9\mathbf{Z}\).
(d.) 142.

17. The number of elements of order 5 in the symmetric group \(S_5\) is

(a.) 5.
(b.) 20.
(c.) 24.
(d.) 12.

18. The order of the element \((\overline{2}, \overline{2})\) in \(\mathbb{Z}_4 \times \mathbb{Z}_6\) is

(a.) 2.
(b.) 6.
(c.) 4.
(d.) 12.

19. If \(H\) be a subgroup of \(\mathbf{Z}_{30}\) then which of the following pairs can be together in \(H\)

(a.) \(\{2, 28\}\).
(b.) \(\{8, 22\}.\)
(c.) \(\{14, 16\}.\)
(d.) None of these.

20. Select the correct statement

(a.) Every finite group has an abelian subgroup.
(b.) Every finite group has a cyclic subgroup.
(c.) Every finite group has a self-inverse element other than the identity.
(d.) None of the above.

21. Which of the following is correct?

(a.) An infinite group can have a unique element of finite order.
(b.) There exists an infinite order group in which the order of every element is finite.
(c.) For every infinite order group \(G\), there exists \(a \in G\) such that \(O(a)\) is finite.
(d.) None of these.

22. Show that the alternating group \(A_n, n \geq 3\) is generated by all cycles of length 3.

(a.) True.
(b.) False.

23. Let \(S_3\) be the group of all permutations on three symbols, with the identity element \(e\). Then the number of elements in \(S_3\) that satisfy the equation \(x^2 = e\) is

(a.) 1.
(b.) 2.
(c.) 3.
(d.) 4.

24. The order of the group generated by the matrices \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\) and \(\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}\), where \(i = \sqrt{-1}\), under matrix multiplication is

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

25. Let \(S_n\) be the group of all permutations on the set \(\{1,2, \ldots, n\}\) under the composition of mappings. For \(n > 2\), if \(H\) is the smallest subgroup of \(S_n\) containing the transposition \((1,2)\) and the cycle \((1,2, \ldots, n)\), then

(a.) \(H = S_n\).
(b.) \(H\) is abelian.
(c.) The index of \(H\) in \(S_n\) is 2.
(d.) \(H\) is cyclic.

26. Let \(G\) be a cyclic group of order 60. Then the number of subgroups of \(G\) is

(a.) 1.
(b.) 2.
(c.) 4.
(d.) 12.

27. Choose the correct statement

(a.) A non-cyclic group can have all of its proper subgroups as cyclic.
(b.) Every finite cyclic group has an even number of generators.
(c.) An infinite cyclic group has precisely two generators.
(d.) Every finite group of composite order possesses a proper subgroup.

28. Let \(G\) be an Abelian group. Let \(H_1 = \{\mathrm{e}, \mathrm{g}_1\}\), \(H_2 = \{\mathrm{e}, \mathrm{g}_2\}\), where \(\mathrm{g}_1 \neq \mathrm{g}_2\), be two subgroups of \(G\), each of order 2.

Assertion (A): \(H = \{\mathrm{e}, \mathrm{g}_1, \mathrm{g}_2, \mathrm{g}_1 \mathrm{g}_2\}\) is a subgroup of \(G\), and \(H\) is not cyclic.

Reason (R): If \(T\) is a finite group of order \(n\) and if \(T\) is cyclic, then \(T\) will contain at least one element \(t\) with \(O(t) = n\) (i.e., \(n\) is the least positive integer with the property that \(t^n = \mathrm{e}\))

(a.) Both A and R are individually true and R is the correct explanation of A.
(b.) Both A and R are individually true but R is not the correct explanation of A.
(c.) A is true but R is false.
(d.) A is false but R is true.

29. Number of subgroups of order 21 in \(Z_6 \oplus Z_{21}\)

(a.) 3.
(b.) 4.
(c.) 6.
(d.) 12.

30. Let \(H < S_n\) such that \(O(H) = 21\). Then

(a.) \(H\) may contain 20 even and one odd permutation.
(b.) \(H\) may contain only even permutations.
(c.) \(H\) may contain only odd permutations.
(d.) \(H\) may contain 20 odd and one even permutation.