Practice Questions for CSIR NET Group Theory : Group Homomorphism II

11. Let \(G=\mathbb{R} \backslash\{0\}\) and \(H=\{-1,1\}\) be groups under multiplication. Then the map \(\varphi: G \rightarrow H\) defined by \(\varphi(x)=\frac{x}{|x|}\) is

(a.) Not a homomorphism
(b.) A one-one homomorphism, which is not onto
(c.) An onto homomorphism, which is not one-one
(d.) An isomorphism

12. Let \(G\) be a cyclic group of order 8, then its group of automorphisms has order

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

13. Let bijections \(f\) and \(g: \mathbb{R} /\{0,1\} \rightarrow \mathbb{R} /\{0,1\}\) be defined by \(f(x)=1 /(1-x)\) and \(g(x)=x /(x-1)\), and let \(G\) be the group generated by \(f\) and \(g\) under composition of mappings. It is given that \(G\) has order 6. Then,

(a.) \(G\) and its automorphisms group are both Abelian
(b.) \(G\) and its automorphisms group are both non-Abelian
(c.) \(G\) is abelian but its automorphisms group is non-abelian
(d.) \(G\) is non-abelian but its automorphisms group is Abelian

14. Let \(G\) and \(H\) be two groups. The groups \(G \times H\) and \(H \times G\) are isomorphic

(a.) For any \(G\) and any \(H\)
(b.) Only if one of them is cyclic
(c.) Only if one of them is abelian
(d.) Only if \(G\) and \(H\) are isomorphic

15. Let \(H=\mathbb{Z}_2 \times \mathbb{Z}_6\) and \(K=\mathbb{Z}_3 \times \mathbb{Z}_4\). Then

(a.) \(H\) is isomorphic to \(K\) since both are cyclic
(b.) \(H\) is not isomorphic to \(K\) since \(K\) is cyclic whereas \(H\) is not
(c.) \(H\) is not isomorphic to \(K\) since there is no homomorphism from \(H\) to \(K\)
(d.) None of these

16. The number of groups of order \(n\) (up to isomorphism) is

(a.) Finite for all values of \(n\)
(b.) Finite only for finitely many values of \(n\)
(c.) Finite for infinitely many values of \(n\)
(d.) Infinite for some values of \(n\)

17. Any subgroup of \(Q\) (the group of rational numbers under addition) is

(a.) Cyclic and finitely generated but not abelian and normal
(b.) Cyclic and abelian but not finitely generated and normal
(c.) Abelian and normal but need not be cyclic and finitely generated
(d.) Finitely generated and normal but not cyclic and abelian

18. Let \(G\) be the additive group of integers \(\mathbb{Z}\) and \(G^{\prime}\) be the multiplicative group of the fourth roots of unity. Let \(f: G \rightarrow G^{\prime}\) be a homomorphism mapping given by \(f(n)=i^n\); where \(i=\sqrt{-1}\). Then the Kernel of \(f\) is

(a.) empty set
(b.) \(\{4 m ; m \in \mathbb{Z}\}\)
(c.) \(\left\{(2 m)^2+1: m \in \mathbb{Z}\right\}\)
(d.) \(\{2 m+1: m \in \mathbb{Z}\}\)

19. Let \(G\) be a group of order 17. The total number of non-isomorphic subgroups of \(G\) is

(a.) 1
(b.) 2
(c.) 3
(d.) 17

20. Let \(G\) be a cyclic group of order 24. The total number of group isomorphisms of \(G\) onto itself is

(a.) 7
(b.) 8
(c.) 17
(d.) 24