Practice Questions for CSIR NET Group Theory : Group Homomorphism III

21. Let \(f\) and \(g\) be the functions from \(\mathbb{R} \backslash\{0,1\}\) to \(\mathbb{R}\) defined by \(f(x)=\frac{1}{x}\) and \(g(x)=\frac{x-1}{x}\) for \(x \in \mathbb{R} \backslash\{0,1\}\). The smallest group of functions from \(\mathbb{R} \backslash\{0,1\}\) to \(\mathbb{R}\) containing \(f\) and \(g\) under composition of functions is isomorphic to

(a.) \(K_4\)
(b.) \(S_3\)
(c.) \(\mathbf{Z}_4\)
(d.) \(Q_8\)

22. Let \(A\) be the group of all rational numbers under addition, \(B\) be the group of all nonzero rational numbers under multiplication and \(C\) the group of all positive rational numbers under multiplication.

(a.) \(A \cong B\)
(b.) \(A \cong C\)
(c.) \(B \cong C\)
(d.) None of these

23. In which of the following pairs are the two groups isomorphic to each other? Justify your answers.

1. \(\mathbf{R} / \mathbb{Z}\) and \(S^1\), where \(\mathbb{R}\) is the additive group of real numbers and \(S^1=\{z \in \mathbb{C}:|z|=1\}\) under complex multiplication.

2. \((\mathbb{Z},+)\) and \((\mathbb{Q},+)\).

(a.) Both are correct
(b.) 1 is correct & 2 is incorrect
(c.) 1 is incorrect & 2 is correct
(d.) Neither 1 nor 2 are correct

24. Let \(G\) be a group of order 7 and \(\phi(x)=x^4\), \(x \in G\). Then \(\phi\)

(a.) not one-one
(b.) not onto
(c.) not a homomorphism
(d.) one-one, onto and homomorphism

25. Let \(S_n\) denote the symmetric group on \(n\) symbols. The group \(S_3 \oplus(\mathbb{Z} / 2 \mathbb{Z})\) is isomorphic to which of the following groups?

(a.) \(\mathbb{Z} / 12 \mathbb{Z}\)
(b.) \((\mathbb{Z} / 6 \mathbb{Z}) \oplus(\mathbb{Z} / 2 \mathbb{Z})\)
(c.) \(A_4\), the alternating group of order 12
(d.) \(D_6\), the dihedral group of order 12

26. The number of distinct group homomorphisms from \(Z_{12}\) to \(Z_{25}\) is

(a.) One
(b.) Two
(c.) Three
(d.) Four

27. Let \(G\) be the group of symmetries of a rhombus. Then \(G\) is isomorphic

(a.) The trivial group \(\{e\}\)
(b.) \(\mathbb{Z}_2\)
(c.) \(\mathbb{Z}_2 \times \mathbb{Z}_2\)
(d.) \(\mathbb{Z}_4\)

28. Which of the following statements is false?

(a.) Every infinite cyclic group is isomorphic to the additive group of all even integers
(b.) All groups of order 4 are isomorphic
(c.) All groups of order 7 are isomorphic
(d.) All non-commutative groups of order 6 are isomorphic

29. Which of the following is a pair of isomorphic group

(a.) \((\mathbb{Z},+)(\mathbb{Q},+)\)
(b.) \((\mathbb{R},+)(\mathbb{C},+)\)
(c.) \(\left(\mathbb{R}^*, \cdot\right)\left(\mathrm{C}^*, \cdot\right)\)
(d.) \((\mathbb{R},+)\left(\mathbb{R}^*, \cdot\right)\)

30. The number of group homomorphisms form \(\mathrm{Z}_2 \times \mathrm{Z}_2\) to \(\mathrm{Z}_4\) is equal to

(a.) 2
(b.) 4
(c.) 6
(d.) 8

31. For an integer \(n \geq 2\), let \(S_n\) be the permutation group on \(n\) letter and \(A_n\) the alternating group. Let \(\mathbb{C}^*\) be the group of non-zero complex numbers under multiplication. Which of the following are correct statements?

(a.) For every integer \(n \geq 2\) there is a nontrivial homomorphism \(\chi: S_n \rightarrow \mathbb{C}^*\).
(b.) For every integer \(n \geq 2\), there is a unique nontrivial homomorphism \(\chi: S_n \rightarrow \mathbb{C}^*\).
(c.) For every integer \(n \geq 3\), there is a nontrivial homomorphism \(\chi: \mathrm{A}_n \rightarrow \mathbb{C}^*\).
(d.) For every integer \(n \geq 5\), there is no nontrivial homomorphism \(\chi: A_n \rightarrow \mathbb{C}^*\).

32. Let \(f: \mathbb{Z} \rightarrow(\mathbb{Z} / 4 \mathbb{Z}) \times(\mathbb{Z} / 6 \mathbb{Z})\) be the function \(f(n)=(n \bmod 4,3 \bmod 6)\). then

(a.) \((0 \bmod 4,3 \bmod 6)\) is in the image of \(f\)
(b.) \((a \bmod 4, b \bmod 6)\) is in the image of \(f\), for all even integers \(a\) and \(b\)
(c.) Image of \(f\) has exactly 6 elements
(d.) Kernel of \(f=24 \mathbb{Z}\)