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

26. Choose the correct:

(a.) Every normal subgroup of a group corresponds to a quotient group.
(b.) Any normal subgroup of group of order 2 is contained in center of group.
(c.) If \( \exists \) only one element of order 2 in any group, then that element will be in center of group.
(d.) All of the above

27. \( G \) is a non-Abelian group such that \( o(G) = 27 \) and then the order of \( Z(G) \) (the center of \( G) \) is:

(a.) 1
(b.) 3
(c.) 9
(d.) 27

28. In the permutation group \( S_{n}(n \geq 5) \), if \( H \) is the smallest subgroup containing all the 3-cycles, then which one of the following is TRUE?

(a.) Order of \( H \) is 2
(b.) Index of \( H \) in \( S_{n} \) is 2
(c.) \( H \) is abelian
(d.) \( H=S_{n} \)

29. Let \( G \) be a non-Abelian group of order \( n \) such that \( n=pq \) where \( p \) and \( q \) are two distinct primes each greater than 2. Then choose incorrect:

(a.) \( Z(G) \) is a trivial subgroup of \( G \)
(b.) There exists \( a \in G \) such that \( |cl(a)|=p \)
(c.) There exists \( a \in G \) such that \( |cl(a)|=q \)
(d.) None of these

30. If \( H \) is a subgroup of \( S_{5} \) that contains all three cycles and \( K=\langle H \cup \{ \sigma \} \rangle \) subgroup generated by where \( n \geq 5 \) and \( \sigma \) is an odd permutation, then:

(a.) Every permutation of \( H \) is an even permutation
(b.) \( H \) is a normal subgroup of \( K \)
(c.) \( H \) is not a normal subgroup of \( S_{5} \)
(d.) \( H \) is a normal subgroup of \( S_{5} \)

31. Determine which of the following cannot be the class equation of a group:

(a.) \(10=1+1+1+2+5\)
(b.) \(4=1+1+2\)
(c.) \(8=1+1+3+3\)
(d.) \(6=1+2+3\)

32. Which of the following are the possible class equation of a group \( G \):

(a.) \(14=1+7+3+3\)
(b.) \(22=1+11+2+2+2+2+2\)
(c.) \(35=1+7+7+5+5+5+5\)
(d.) \(21=1+7+7+3+3\)

33. Let \( G \) be a finite group and \( H \) be a normal subgroup of \( G \) of order 2. Then the order of the center of \( G \) is:

(a.) 0
(b.) 1
(c.) an even integer \( \geq 2 \)
(d.) an odd integer \( \geq 3 \)

34. Suppose \( N \) is a normal subgroup of a group \( G \). Which one of the following is true?

(a.) If \( G \) is an infinite group then \( G / N \) is an infinite group
(b.) If \( G \) is a nonabelian group then \( G / N \) is not a nonabelian group
(c.) If \( G \) is a cyclic group then \( G / N \) is an abelian group
(d.) If \( G \) is an abelian group then \( G / N \) is a cyclic group

35. The number of \( 2 \times 2 \) matrices over \( \mathbb{Z}_{3} \) (the field with three elements) with determinant 1 is:

(a.) 24
(b.) 60
(c.) 20
(d.) 30

36. Pick out the cases where the given subgroup \( H \) is not a normal subgroup of the group \( G \)

(a.) \( G \) is the group of all \( 2 \times 2 \) invertible upper triangular matrices with real entries, under matrix multiplication, and \( H \) is the subgroup of all such matrices \( [a_{ij}] \) such that \( a_{11}=1 \)
(b.) \( G \) is the group of all \( 2 \times 2 \) invertible upper triangular matrices with real entries, under matrix multiplication, and \( H \) is the subgroup of all such matrices \( [a_{ij}] \) such that \( a_{11}=a_{22} \)
(c.) \( G \) is the group of all \( 2 \times 2 \) invertible upper triangular matrices with real entries, under matrix multiplication, and \( H \) is the subgroup of all such matrices \( [a_{ij}] \) with positive determinant
(d.) All of the above

37. Let \( G \) be a group of order 49. Then:

(a.) \( G \) is abelian
(b.) \( G \) is cyclic
(c.) \( G \) is non-abelian
(d.) Center of \( G \) has order 7

38. Consider the quotient group \( \mathbb{Q} / \mathbb{Z} \) of the additive group of rational numbers. The order of the element \( \frac{2}{3}+\mathbb{Z} \) in \( \mathbb{Q} / \mathbb{Z} \) is:

(a.) 2
(b.) 3
(c.) 5
(d.) 6

39. Let \( G \) denote the group of all \( 2 \times 2 \) invertible matrices with entries from \( \mathbb{R} \). Let \( H_{1}=\{A \in G: \operatorname{det}(A)=1\} \) and \( H_{2}=\{A \in G: A \) is upper triangular \( \} \). Consider the following statements:

\( P: H_{1} \) is a normal subgroup of \( G \)

\( Q: H_{2} \) is a normal subgroup of \( G \).

(a.) Both \( P \) and \( Q \) are true
(b.) \( P \) is true and \( Q \) is false
(c.) \( P \) is false and \( Q \) is true
(d.) Both \( P \) and \( Q \) are false

40. Consider the following statements \( P \) and \( Q \):

\( [P]: \) If \( H \) is a normal subgroup of order 4 of the symmetric group \( S_{4} \), then \( S_{4} / H \) is Abelian.

\( [Q]: \) If \( Q=\{ \pm 1, \pm i, \pm j, \pm k\} \) is the quaternion group, then \( Q /\{-1,1\} \) is Abelian.

Which of the above statements hold TRUE?

(a.) Both \( P \) and \( Q \)
(b.) Only \( P \)
(c.) Only \( Q \)
(d.) Neither \( P \) and \( Q \)