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

11. Let \( p \) be a prime number. The order of \( p- \)Sylow subgroup of the group \( GL(50, F_{p}) \) of invertible \( 50 \times 50 \) matrices with entries from the finite field \( \mathbb{F}_{p} \), equals:

(a.) \( p^{50} \)
(b.) \( p^{125} \)
(c.) \( p^{1250} \)
(d.) \( p^{1225} \)

12. Let \( G \) be a group of order 56. Then:

(a.) All 7-Sylow subgroups of \( G \) are normal
(b.) All 2-Sylow subgroups of \( G \) are normal
(c.) Either a 7-Sylow subgroup or a 2-Sylow subgroup of \( G \) is normal
(d.) There is a proper normal subgroup of \( G \)

13. Up to an isomorphism, the number of groups of order 33 is:

(a.) 3
(b.) 11
(c.) 1
(d.) Infinitely many

14. The number of 5-Sylow subgroups of \( S_{6} \) is:

(a.) 16
(b.) 6
(c.) 36
(d.) 1

15. Let \( G \) be a group of order 14 such that \( G \) is not abelian. Then the number of elements of order 2 in \( G \) is equal to:

(a.) 7
(b.) 6
(c.) 13
(d.) 2

16. The number of groups of order 121, up to isomorphism, is:

(a.) 1
(b.) 2
(c.) 11
(d.) 10

17. Let \( G \) be a non-abelian group of order 21. Let \( H \) be a Sylow 3-subgroup and \( K \) be a Sylow 7-subgroup of \( G \). Then:

(a.) \( H \) and \( K \) are both normal in \( G \)
(b.) \( H \) is normal but \( K \) is not normal in \( G \)
(c.) \( K \) is normal but \( H \) is not normal in \( G \)
(d.) Neither \( H \) nor \( K \) is normal in \( G \)

18. If \( H \) and \( K \) are subgroup of \( G \) with indices 3 and 5 in \( G \), then the index of \( H \cap K \) in \( G \) is:

(a.) 3
(b.) 5
(c.) a multiple of 15
(d.) not more than 8

19. The number of 5-Sylow subgroup(s) in a group of order 45 is:

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

20. Let \( \omega = \cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3} \), \( M = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} \), \( N = \begin{pmatrix} \omega & 0 \\ 0 & \omega^{2} \end{pmatrix} \), and \( G = \langle M, N \rangle \) be the group generated by the matrices \( M \) and \( N \) under matrix multiplication. Then:

(a.) \( G / Z(G) \cong \mathbb{Z}_{6} \)
(b.) \( G / Z(G) \cong S_{3} \)
(c.) \( G / Z(G) \cong \mathbb{Z}_{2} \)
(d.) \( G / Z(G) \cong \mathbb{Z}_{4} \)