B.Sc. Mathematics Honours C-6 Question Paper 2018 (CBCS)

Group Theory-I

Unit-I

1. Answer any two questions : $2 \times 2$
(a) Is the set $R^{*}$ of all non-zero real numbers a group with respect to the operations $\circ$ defined by $a \circ b = |a|b$ for all $a, b \in R^{*}$? Justify your answer.
(b) Let $(G, *)$ be a group of even order. Show that there exists $a \in G$ such that $a \neq e, a^{2} = e$.
(c) Let $(G, \circ)$ be a group. Define a mapping $f: G \rightarrow G$ by $f(x) = x^{-1}, x \in G$. Prove that $f$ is a bijection.

2. Answer any one question : $1 \times 5$
(a) Show that the set of six transformations $f_{1}, f_{2}, f_{3}, f_{4}, f_{5}$ and $f_{6}$ on the set of complex numbers defined by $f_{1}(z) = z, f_{2}(z) = \frac{1}{z}, f_{3}(z) = 1 - z, f_{4}(z) = \frac{z}{z-1}, f_{5}(z) = \frac{1}{1-z}$ and $f_{6}(z) = \frac{z-1}{z}$ forms a finite non-Abelian group of order 6 with respect to the composition of mapping.
(b) Construct the dihedral group $D_{4}$ from the symmetries of a square. Show that the order of it is 8.

Unit-II

3. Answer any two questions : $2 \times 2$
(a) A non-Abelian group can have an Abelian subgroup. Justify the statement with example.
(b) In a group $(G, \bullet), (ab)^{3} = a^{3}b^{3} \forall a, b \in G$. Show that $H = \{x^{3} : x \in G\}$ is a subgroup of $G$.
(c) Let $(G, \circ)$ be a group and $H, K$ are subgroups of $(G, \circ)$. Then show that $H \cap K$ is a subgroup of $(G, \circ)$.

4. Answer any two questions : $2 \times 5$
(a) Prove that $H$ is a subgroup of $Z_{12}$ where $H = \{\overline{0}, \overline{2}, \overline{4}, \overline{6}, \overline{8}, \overline{10}\}$.
(b) Let $H, K$ be subgroups of a group $G$. Prove that set $HK$ is a subgroup of $G$ iff $HK = KH$, where $HK = \{hk : h \in H \text{ and } k \in K\}$ and $KH = \{kh : k \in K \text{ and } h \in H\}$.
(c) Let $H$ be a subgroup of a group $G$ and $a \in G$. Define normalizer of $a$ in $G$ and centralizer of $H$ in $G$. Show that centralizer of $H$ and normalizer of $H$ in $G$ are not same. Justify your answer with example.

Unit-III

5. Answer any two questions : $2 \times 2$
(a) Let $G$ be a finite group, $A$ and $B$ be two subgroups of $G$ such that $A \subseteq B \subseteq G$. Prove that, $[G:A] = [G:B][B:A]$.
(b) Show that a cyclic group with only one generator can have at most two elements.
(c) Determine all distinct left cosets of $A_{3}$ in $S_{3}$.

6. Answer any one question : $1 \times 10$
(a) (i) Let $H$ be a subgroup of a group $G$. Then show that the set of all distinct left cosets of $H$ in $G$ have the same cardinality.
(ii) Show that the number of even permutation of a finite set (containing at least two elements) is equal to the number of odd permutation on it. $5+5$
(b) Prove that, a finite group of order $n$ is cyclic if and only if it has an element of order $n$. Also prove that every subgroup of a cyclic group is cyclic. $5+5$

Unit-IV

7. Answer any two questions : $2 \times 2$
(a) Let $G = \{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} : a, b, c \text{ are real and } ac \neq 0 \}$ be a group under matrix multiplication. Show that $N = \{ \begin{pmatrix} 1 & c \\ 0 & 1 \end{pmatrix} : c \text{ is a real number} \}$ is a normal subgroup of $G$.
(b) If $H$ be a subgroup of a commutative group $G$ then show that $G/H$ is commutative.
(c) Show that if $p$ is a prime number, then any group $G$ of order $2p$ has a normal subgroup of order $p$.

8. Answer any one question : $1 \times 10$
(a) Define centre $Z(G)$ of a group $G$. Prove that $Z(G)$ is a normal subgroup of $(G, \circ)$. Also prove that $mn = nm \forall m \in M \text{ and } n \in N$, where $M$ and $N$ are two normal subgroups of a group $G$. Show that $M \cap N = \{e\}$, $e$ being the identity element in $G$. $2+4+4$
(b) State and prove Cauchy's theorem for finite Abelian groups. $2+8$

Unit-V

9. Answer any two questions : $2 \times 2$
(a) Define $f: (S_{3}, \circ) \rightarrow (\{1, -1\}, \bullet)$ by $f(\alpha) = 1$, if $\alpha$ is an even permutation in $S_{3}$ and $f(\alpha) = -1$, if $\alpha$ is an odd permutation in $S_{3}$. Show that $f$ is homomorphism from $(S_{3}, \circ)$ to $(\{1, -1\}, \bullet)$, $\circ$ is the composition of mapping.
(b) Show that $Ker \ f$ (Kernel of homomorphism) from $(G, \circ)$ to $(G', *)$ is a normal subgroup of $G$.
(c) Let $GL(2,R)$ be the group of non-singular real matrices under multiplication, $R^{*}$ be the group of non-zero reals under multiplication and a function $F: GL(2,R) \rightarrow R^{*}$ is defined by $f(\begin{bmatrix} a & b \\ c & d \end{bmatrix}) = ad - bc$. Show that $f$ is a homomorphism.

10. Answer any one question : $1 \times 5$
(a) Prove that every finite group $G$ is isomorphic to a permutation group.
(b) If $H$ and $K$ are two normal subgroup of $G$ such that $H \subseteq K$, then show that $\frac{G}{K} \cong \frac{G/H}{K/H}$.