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

1. For a positive integer \( n \geq 4 \) and a prime number \( p \leq n \), let \( U_{p, n} \) denote the union of all \( p \)-Sylow subgroups of the alternating group \( A_{n} \) on \( n \) letters. Also let \( K_{p, n} \) denote the subgroup of \( A_{n} \) generated by \( U_{p, n} \), and let \( \left|K_{p, n}\right| \) denote the order of \( K_{p, n} \). Then:

(a.) \( \left|K_{2,4}\right|=12 \)
(b.) \( \left|K_{2,4}\right|=4 \)
(c.) \( \left|K_{2,5}\right|=60 \)
(d.) \( \left|K_{3,5}\right|=30 \)

2. Let \( G \) be a group of order 231. The number of elements of order 11 in \( G \) is:

(a.) 1
(b.) 5
(c.) 10
(d.) 20

3. Up to isomorphism, the number of abelian groups of order 108 is:

(a.) 12
(b.) 9
(c.) 6
(d.) 5

4. In the group of all invertible \( 4 \times 4 \) matrices with entries in the field of 3 elements, any 3-Sylow subgroup has cardinality:

(a.) 3
(b.) 81
(c.) 243
(d.) 729

5. Let \( G \) be a group of order 45. Then:

(a.) \( G \) has an element of order 9
(b.) \( G \) has a subgroup of order 9
(c.) \( G \) has a normal subgroup of order 9
(d.) \( G \) has a normal subgroup of order 5

6. Let \( G \) be a non-abelian group. Then its order can be:

(a.) 25
(b.) 55
(c.) 125
(d.) 35

7. The total number of non-isomorphic groups of order 122 is:

(a.) 2
(b.) 1
(c.) 61
(d.) 4

8. How many normal subgroups does a nonabelian group \( G \) of order 21 have other than the identity subgroup \( \{e\} \) and \( G \) ?

(a.) 0
(b.) 1
(c.) 3
(d.) 7

9. Let \( G \) be a simple group of order 168. What is the number of subgroups of \( G \) of order 7 ?

(a.) 1
(b.) 7
(c.) 8
(d.) 28

10. Let \( G \) denote the group \( S_{4} \times S_{3} \). Then:

(a.) a 2-Sylow subgroup of \( G \) is normal
(b.) a 3-Sylow subgroup of \( G \) is normal
(c.) \( G \) has a nontrivial normal subgroup
(d.) \( G \) has a normal subgroup of order 72