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

1. Which of the following cannot be the class equation of a group of order 10?

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

2. Let \(G\) be the group of all symmetries of the square. Then the number of conjugate classes in \(G\) is

(a.) 4
(b.) 5
(c.) 6
(d.) 7

3. 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

4. The number of conjugacy classes in the permutation group \(S_{6}\) is:

(a.) 12
(b.) 11
(c.) 10
(d.) 6

5. 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.) \(6=1+2+3\)

6. Let

\(\sigma=\left(\begin{array}{ll} 1 & 2 \end{array}\right)\left(\begin{array}{lll} 3 & 4 & 5 \end{array}\right)\)

and \(\tau=\left(\begin{array}{llllll}1 & 2 & 3 & 4 & 5 & 6\end{array}\right)\) be permutations in \(S_{6}\), the group of permutations on six symbols. Which of the following statements are true?

(a.) The subgroups \(\langle\sigma\rangle\) and \(\langle\tau\rangle\) are isomorphic to each other
(b.) \(\sigma\) and \(\tau\) are conjugate in \(S_{6}\)
(c.) \(\langle\sigma\rangle \cap\langle\tau\rangle\) is the trivial group
(d.) \(\sigma\) and \(\tau\) commute

7. Consider the group \(G=\mathbb{Q} / \mathbb{Z}\) where \(\mathbb{Q}\) and \(\mathbb{Z}\) are the groups of rational numbers and integers respectively. Let \(n\) be a positive integer. Then is there a cyclic subgroup of order \(n\)?

(a.) Not necessarily exists
(b.) Always exist and a unique one
(c.) Always exist and, but not necessarily a unique one
(d.) never exist

8. For any group \(G\) of order 36 and any subgroup \(H\) of \(G\) order 4

(a.) \(H \subset Z(G)\)
(b.) \(H=Z(G)\)
(c.) \(H\) is normal in \(G\)
(d.) \(H\) is an abelian group

9. Consider a group \(G\). Let \(Z(G)\) be its centre, i.e. \(Z(G)=\{g \in G: g h=h g\) for all \(h \in G\}\). For \(n \in \mathbb{N}\), the set of positive integers, define \(J_{n}=\left\{\left(g_{1}, \ldots, g_{n}\right) \in Z(G) \times \ldots . . \times Z(G): g_{1} \ldots g_{n}=e\right\}\). As a subset of the direct product group \(G \times \ldots \times G\) (\(n\) times direct product of the group \(G\)), \(J_{n}\) is

(a.) Not necessarily a subgroup
(b.) A subgroup but not necessarily a normal subgroup
(c.) A normal subgroup
(d.) Isomorphic to the direct product \(Z(G) \times \ldots . . \times Z(G)\) \((n-1)\) times

10. Let \(G\) be a group of order 77. Then the center of \(G\) is isomorphic to

(a.) Trivial group
(b.) \(\mathbb{Z}_{7}\)
(c.) \(\mathbb{Z}_{11}\)
(d.) \(\mathbb{Z}_{77}\)

11. Let \(H=\{e,(1,2)(3,4)\}\) and \(K=\{e,(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}\) be subgroups of \(S_{4}\), where \(e\) denotes the identity element of \(S_{4}\). Then

(a.) \(H\) and \(K\) are normal subgroups of \(S_{4}\)
(b.) \(H\) is normal in \(K\) and \(K\) is normal in \(A_{4}\)
(c.) \(H\) is normal in \(A_{4}\) but not normal in \(S_{4}\) unique normal subgroup of order four?
(d.) \(K\) is normal in \(S_{4}\), but \(H\) is not.

12. \(S_{n}=\) Symmetric group of permutations of \(n\) symbols and \(A_{n}=\) Set of all even permutations on \(n\)-symbols then

(a.) \(A_{n}\) is subgroup of \(S_{n}\) but not a normal subgroup of \(S_{n}\).
(b.) \(A_{n}\) is normal subgroup of \(S_{n}\) and \(\frac{S_{n}}{A_{n}}\) is cyclic.
(c.) \(A_{n}\) is not a subgroup of \(S_{n}\).
(d.) None of these