Practice Questions for CSIR NET Group Theory : Group Homomorphism I

1. The number of group homomorphisms from the symmetric group \(S_3\) to the additive group \(\mathbb{Z} / 6 \mathbb{Z}\) is

(a.) 1
(b.) 2
(c.) 3
(d.) 0

2. Consider the two statements:

1. \(\mathbb{Z}_6\) is a subgroup of \(\mathbb{Z}_{12}\)

2. \(\mathbb{Z}_5\) is isomorphic to a subgroup of \(\mathbb{Z}_{15}\)

Then

(a.) 1 is correct, 2 is incorrect.
(b.) 1 is incorrect, 2 is correct.
(c.) Both are correct.
(d.) Both are incorrect.

3. The number of homomorphisms from \(Q_8\) to \(\mathbb{Z}_8\)

(a.) 4
(b.) 3
(c.) 1
(d.) None of these

4. The number of one-one homomorphisms from \(\mathbb{Z}\) to \(S_{10}\) are

(a.) 10
(b.) 10 !
(c.) 2
(d.) None of these

5. The group of one-one homomorphisms from \(\mathbf{Z}_{21}\) to \(\mathbb{Z}_3 \times \mathbb{Z}_7\) is isomorphic to

(a.) \(\mathbf{Z}_{12}\)
(b.) \(U(12)\)
(c.) \(\mathbb{Z}_{21}\)
(d.) \(\mathbb{Z}_2 \times \mathbb{Z}_6\)

6. The number of group homomorphisms from the symmetric group \(S_3\) to \(\mathbb{Z}_6\) is

(a.) 1
(b.) 2
(c.) 3
(d.) 6

7. The number of non-isomorphic groups of order 10 is

(a.) 1
(b.) 2
(c.) 3
(d.) 4

8. Consider the group homomorphism \(\varphi: M_2(\mathbb{R}) \rightarrow \mathbb{R}\) given by \(\varphi(A)=\text{trace}(A)\). The kernel of \(\varphi\) is isomorphic to which of the following groups?

(a.) \(\frac{M_2(\mathbb{R})}{H}\), where \(H=\left\{A \in M_2(\mathbb{R}): \varphi(A)=0\right\}\)
(b.) \(\mathbb{R}^2\)
(c.) \(\mathbb{R}^3\)
(d.) \(GL(2, \mathbb{R})\)

9. The number of group homomorphisms from \(\mathbb{Z}_3\) to \(\mathbb{Z}_9\) is

(a.) 1
(b.) 2
(c.) 3
(d.) 4

10. Let \(\text{Aut}(G)\) denote the group of automorphisms of a group \(G\). Which one of the following is NOT a cyclic group?

(a.) \(\text{Aut}(Z_4)\)
(b.) \(\text{Aut}(Z_6)\)
(c.) \(\text{Aut}(Z_8)\)
(d.) \(\text{Aut}(Z_{10})\)