Practice Questions for CSIR NET Group Theory : Groups and Their Properties III

31. Consider the group \(G = \mathbf{Z}_4 \times \mathbf{Z}_4\) of order 16, let \(U_k\) be the union of subgroups of order \(k\), and \(H_k = \left\langle U_k \right\rangle\) be a subgroup of \(G\) for each \(k\).

(a.) \(H_4\) is a proper subgroup of \(G\).
(b.) \(H_2\) is a proper subgroup of \(G\).
(c.) \(H_4\) is a cyclic subgroup of \(G\).
(d.) \(H_2\) is a cyclic subgroup of \(G\).

32. Let \(S_{10}\) be a group of permutations of 10 symbols such that \(\beta \in S_{10}\) is a 10-cycle and \(\alpha = \beta^k\), where \(2 \leq k \leq 10\). Then the value of \(k\) for which \(\alpha\) and \(\beta\) have the same order is

(a.) 3.
(b.) 4.
(c.) 5.
(d.) 6.

33. Let \(G\) be a finite group and \(o(G)\) denotes its order. Then which of the following statement(s) is(are) TRUE?

(a.) \(G\) is abelian if \(o(G) = p q\) where \(p\) and \(q\) are distinct primes.
(b.) \(G\) is abelian if every non-identity element of \(G\) is of order 2.
(c.) \(G\) is abelian if the quotient group \(\frac{G}{Z(G)}\) is cyclic, where \(Z(G)\) is the center of \(G\).
(d.) \(G\) is abelian if \(o(G) = p^3\), where \(p\) is prime.

34. Let \(S\) be the group of permutations of three distinct symbols. The direct sum \(S_3 \oplus S_3\) has an element of order

(a.) 2.
(b.) 6.
(c.) 9.
(d.) 18.

35. Which of the following statements is (are) true?

(a.) \(Z_2 \oplus Z_3\) is isomorphic to \(Z_6\).
(b.) \(Z_2 \oplus Z_4\) is isomorphic to \(Z_8\).
(c.) \(Z_4 \oplus Z_6\) is isomorphic to \(Z_{24}\).
(d.) \(Z_2 \oplus Z_3 \oplus Z_5\) is isomorphic to \(Z_{30}\).

36. Let \(G\) be a non-abelian group and \(a, \beta \in G\) have order 3, 3 respectively. Then the order of the element \(\alpha \beta\) in \(G\) is

(a.) 156.
(b.) Is 12.
(c.) Is of the form \(12k\) for \(k \geq 2\).
(d.) Need not be finite.

37. Consider the following statements:

1. \(A(S)\) be a group of bijections on a non-empty set. Then for any \(x_0 \in S\), \(H = \{\phi \in A(S): \phi(x_0) = x_0\}\) is a subgroup of \(A(S\).

2. \(G\) be a group of mappings \(f_{a, b}: R \rightarrow R\) defined by \(f_{a, b}(x) = ax + b\) where \(a \neq 0\). Then \(H = \{f_{a} \in G: a\) is a rational number\} is a subgroup of \(G\).

(a.) I correct, II incorrect.
(b.) II correct, I incorrect.
(c.) Both I and II are correct.
(d.) Both I and II are incorrect.

38. The number of elements of \(S_3\) (the symmetric group on 5 letters) which are their own inverses equals

(a.) 10.
(b.) 11.
(c.) 25.
(d.) 26.

39. Which one of the following conditions on a group \(G\) implies that \(G\) is abelian?

(a.) The order of \(G\) is \(p^3\) for some prime \(p\).
(b.) Every proper subgroup of \(G\) is cyclic.
(c.) Every subgroup of \(G\) is normal in \(G\).
(d.) The function \(f: G \rightarrow G\), defined by \(f(x) = x^{-1}\) for all \(x \in G\), is a homomorphism.

40. Let \(p\) be a prime number. Let \(G\) be the group of all \(2 \times 2\) matrices over \(Z_p\) with determinant 1 under matrix multiplication. Then the order of \(G\) is

(a.) \((p-1) p(p+1)\).
(b.) \(p^2(p-1)\).
(c.) \(p^3\).
(d.) \(p^2(p-1) + p\).

41. Let \(G\) be a group of order 125. Which of the following statements are necessarily true?

(a.) \(G\) has a non-trivial abelian subgroup.
(b.) The center of \(G\) is a proper subgroup.
(c.) The center of \(G\) has order 5.
(d.) There is a subgroup of order 25.

42. Let \(G\) be a finite abelian group and \(a, b \in G\) with \(\text{order}(a) = m\), \(\text{order}(b) = n\). Which of the following are necessarily true?

(a.) \(\text{Order}(ab) = mn\).
(b.) \(\text{Order}(ab) = 1\ \text{cm}(m, n)\).
(c.) There is an element of \(G\) whose order is \(\text{lcm}(m, n)\).
(d.) \(\text{Order}(ab) = \text{gcd}(m, n)\).