Practice Questions for CSIR NET Group Theory : Sylows' Theorems and Their Applications III

21. If \( Z(G) \) denotes the center of a group \( G \), then the order of the quotient group \( \frac{G}{Z(G)} \) cannot be:

(a.) 4
(b.) 6
(c.) 15
(d.) 25

22. Let \( G \) be a group of order 45. Let \( H \) be a 3-Sylow subgroup of \( G \) and \( K \) be a 5-Sylow subgroup of \( G \). Then:

(a.) Both \( H \) and \( K \) are normal in \( G \)
(b.) \( H \) is normal in \( G \) but \( K \) is not normal in \( G \)
(c.) \( H \) is not normal in \( G \) but \( K \) is normal in \( G \)
(d.) Both \( H \) and \( K \) are not normal in \( G \)

23. Let \( G \) be the group with the generators \( a \) and \( b \) given by \( G = \langle a, b: a^{4} = b^{2} = e, ba = a^{-1}b \rangle \). If \( Z(G) \) denotes the center of \( G \), then \( \frac{G}{Z(G)} \) is isomorphic to:

(a.) The trivial group
(b.) \( \mathbb{Z}_{2} \)
(c.) \( \mathbb{Z}_{2} \times \mathbb{Z}_{2} \)
(d.) \( \mathbb{Z}_{4} \)

24. The number of 5-Sylow subgroups of \( \mathbb{Z}_{20} \) is:

(a.) 1
(b.) 4
(c.) 5
(d.) 6

25. Up to isomorphism, the number of abelian groups of order \(10^{5}\) is:

(a.) 2
(b.) 5
(c.) 7
(d.) 49

26. Let \(G\) be a finite group of order 200. Then the number of subgroups of \(G\) of order 25 is:

(a.) 1
(b.) 4
(c.) 5
(d.) 10

27. Which of the following is false:

(a.) Any abelian group of order 27 is cyclic.
(b.) Any abelian group of order 14 is cyclic.
(c.) Any abelian group of order 21 is cyclic.
(d.) Any abelian group of order 30 is cyclic.

28. For which of the following integers \(n\) is every group of order \(n\) abelian?

(a.) \(n=6\)
(b.) \(n=9\)
(c.) \(n=12\)
(d.) \(n=18\)

29. Let \(G\) be a group of order 121. Then:

(a.) \(G\) must be cyclic
(b.) \(G\) must have an element of order 11
(c.) \(G\) must have an element of order 121
(d.) \(G\) cannot have an element of order 11

30. The number of mutually non-isomorphic groups of order 45 is:

(a.) 0
(b.) 1
(c.) 2
(d.) 4

31. Consider a group \(G\) of order 21, choose the correct:

(a.) If \(G\) has a unique subgroup of order 3 then \(G\) is cyclic
(b.) If \(G\) is non-cyclic, then there are 14 elements of order 3
(c.) If \(G\) has exactly two elements of order 3 then \(\forall a \in G, N(a)=G\) (the normalizer)
(d.) Converse of Lagrange's theorem holds for group \(G\)

32. Let \(G\) be a group of order 159 and let \(H=\left\{x \in G \mid x^{53}=e\right\}\) then:

(a.) \(H\) is a normal subgroup of \(G\)
(b.) \(G\) has a unique subgroup of order 3
(c.) The number of quotient groups of \(G\) up to isomorphism is 4
(d.) \(G\) is not a simple group

33. Let \(G\) be a group such that \(|G| = 60\). Then:

(a.) \(G\) is abelian
(b.) \(G\) has subgroups of order 2, 3, 5
(c.) \(G\) has subgroups of order 4, 6, 15
(d.) \(G\) has subgroups of order 20, 30

34. Let \(G\) be the set of all abelian groups of order 900. The smallest cardinality of a subset of \(G\), which always contains at least two isomorphic groups, is:

(a.) 6
(b.) 7
(c.) 8
(d.) 9

35. Let \( G \) be a group of order 15. Then the number of Sylow subgroups of \( G \) of order 3 is:

(a.) 0
(b.) 1
(c.) 3
(d.) 5