Practice Questions for CSIR NET Group Theory : Conjugate Classes and Class Equation II

Practice Questions for NET JRF Group Theory
Assignment: Conjugate Classes and Class Equation

13. A finitely generated subgroup \(G\) of the additive group \(\mathbb{Q}\) of rational numbers satisfies

(a.) \(G\) is isomorphic to \(\mathbb{Q}\)
(b.) \(G\) is trivial group
(c.) \(G\) is cyclic
(d.) \(G\) is finite.

14. In the permutation group \(S_{6}\) then numbers of elements conjugate to \((123)(456)\) is

(a.) 80
(b.) 120
(c.) 60
(d.) 40

15. Let \(G=\{e,x,x^2,y,xy,x^2y,x^3y\}\) with \(o(x)=4,o(y)=2\) and \(xy=yx^3\) Then the number of elements in center of the group \(G\) is

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

16. Which of the following groups contains a

(a.) \(\mathbb{Z}_{2} \oplus \mathbb{Z}_{4}\)
(b.) The dihedral group, \(D_{4}\), of order eight
(c.) The quaternion group, \(Q_{8}\)
(d.) None of these

17. The number of elements in the conjugacy class of the 3-cycle (234) in the symmetric group \(S_{6}\) is

(a.) 20
(b.) 40
(c.) 120
(d.) 216

18. Which one of the following groups is simple?

(a.) \(S_{3}\)
(b.) \(G L(2, \mathbb{R})\)
(c.) \(\mathbb{Z}_{2} \times \mathbb{Z}_{2}\)
(d.) \(A_{5}\)

19. Let \(G\) be the group of all symmetries of the square. Then the number of conjugate classes in \(G\) is

(a.) 4
(b.) 5
(c.) 6
(d.) 7

20. Let \(S_{10}\) denote the group of permutations on ten symbols \(\{1,2, \ldots \ldots, 10\}\). The number of elements of \(S_{10}\) commuting with the element \(\sigma=(13579)\) is

(a.) 5 !
(b.) 5.5 !
(c.) \(5 ! \times 5 !\)
(d.) \(\frac{10 !}{5 !}\)

21. Let \(\sigma\) and \(\tau\) be the permutations defined by

\(\sigma=\left(\begin{array}{lllllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 1 & 3 & 5 & 7 & 9 & 6 & 4 & 8 & 2\end{array}\right)\) and

\(\tau=\left(\begin{array}{lllllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 7 & 8 & 3 & 4 & 9 & 6 & 5 & 2 & 1\end{array}\right)\)

Then

(a.) \(\sigma\) and \(\tau\) generate the group of permutation on \(\{1,2,3,4,5,6,7,8,9\}\)
(b.) \(\sigma\) is Contained in the group generated by \(\tau\)
(c.) \(\tau\) is contained in the group generated by \(\sigma\)
(d.) \(\sigma\) and \(\tau\) are in the same conjugacy class

22. Let \(G\) be a group of order 30. Let \(H\) and \(K\) be normal subgroups of orders 2 and 5 respectively. Then the order of the group \(G / (H K)\) is

(a.) 10
(b.) 3
(c.) 2
(d.) 5

23. The number of elements of \(S_{5}\) that commutes with \((123)\) are

(a.) 20
(b.) Equal to self conjugate elements of \(\mathbb{Z}_{6}\)
(c.) 5 !
(d.) None of the above

24. The possible class equation of group of order 10

(a.) \(1+1+1+1+1+5\)
(b.) \(1+2+2+5\)
(c.) \(5+5\)
(d.) \(1+1+1+1+1+1+2+2\)

25. Choose the correct:

(a.) \(\mathbb{Z} / n \mathbb{Z}\) is a cyclic group of order \(n\)
(b.) \(\mathbb{R} / n \mathbb{R}\) is a cyclic group of order \(n\)
(c.) Every factor group of an abelian group is abelian
(d.) All of the above

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