Course Code: MTE-06

Assignment Code: MTE-06/TMA/2023

Maximum Marks: 100

1. Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so.)

i) \(\phi(\mathrm{n})=\mathrm{n}-1 \forall \mathrm{n} \in \mathbb{N}\), where \(\phi\) is the Euler-phi function.

ii) If \(\mathrm{G}_{1}\) and \(\mathrm{G}_{2}\) are groups, and \(\mathrm{f}: \mathrm{G}_{1} \rightarrow \mathrm{G}_{2}\) is a group homomorphism, then \(o\left(G_{1}\right)=o\left(G_{2}\right)\).

iii) If \(\mathrm{G}\) is an abelian group, then \(\mathrm{G}\) is cyclic.

iv) If \(\mathrm{G}\) is a group and \(H \Delta \mathrm{G}\), then \(|\mathrm{G}: \mathrm{H}|=2\).

v) Every element of \(S_{n}\) has order at most \(n\).

vi) If \(R\) is a ring and \(I\) is an ideal of \(R\), then \(x r=r x \forall x \in I\) and \(r \in R\).

vii) If \(\sigma \in \mathrm{S}_{\mathrm{n}}(\mathrm{n} \geq 3)\) is a product of an even number of disjoint cycles, then \(\operatorname{sign}(\sigma)=1\).

viii) If a ring has a unit, then it has only one unit.

ix) The characteristic of a finite field is zero.

x) The set of discontinuous functions from \([0,1]\) to \(\mathbb{R}\) form a ring with respect to pointwise addition and multiplication.

2. a) Define a relation \(\mathrm{R}\) on \(\mathbb{Z}\), by \(\mathrm{R}=\{(\mathrm{n}, \mathrm{n}+3 \mathrm{k}) \mid \mathrm{k} \in \mathbb{Z}\}\).

Check whether \(\mathrm{R}\) is an equivalence relation or not. If it is, find all the distinct equivalence classes. If \(\mathrm{R}\) is not an equivalence relation, define an equivalence relation on \(\mathbb{Z}\).

b) Consider the set \(X=\mathbb{R} \backslash\{-1\}\). Define \(*\) on \(X\) by \(\mathrm{x}_{1} * \mathrm{x}_{2}=\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{1} \mathrm{x}_{2} \forall \mathrm{x}_{1}, \mathrm{x}_{2} \in \mathrm{X}\).

i) Check whether \((\mathrm{X}, *)\) is a group or not.

ii) Prove that \(\mathrm{x} * \mathrm{x} * \mathrm{x} * \ldots * \mathrm{x}\left(\mathrm{n}\right.\) times) \(=(1+\mathrm{x})^{\mathrm{n}}-1 \forall \mathrm{n} \in \mathbb{N}\) and \(\mathrm{x} \in \mathrm{X}\).

c) Give an example, with justification, of a commutative subgroup of a noncommutative group.

3. a) Check whether or not \(\mathrm{A}=\left\{\mathrm{z} \in \mathbb{C}^{*}|| \mathrm{z} k \mathbb{Q}\right\}\) is a subgroup of
i) \(\left(\mathbb{C}^{*},.\right)\)
ii) \((\mathbb{C},+)\)

b) Let \((\mathrm{G}\), .) be a finite abelian group and \(m \in \mathbb{N}\). Prove that \(\mathrm{S}=\{\mathrm{g} \in \mathrm{G} \mid(\mathrm{o}(\mathrm{g}), \mathrm{m})=1\} \leq \mathrm{G}\) c) Let \(G\) be a group of order \(n \geq 2\), with only two subgroups \(-\{\mathrm{e}\}\) and itself. Find a minimal generating set for \(\mathrm{G}\). Also, find out whether \(\mathrm{n}\) is a prime or a composite number, or can be either.

4. a) Consider the map \(\mathrm{f}_{\mathrm{ab}}: \mathbb{R} \rightarrow \mathbb{R}: \mathrm{f}_{\mathrm{ab}}(\mathrm{x})=\mathrm{ax}+\mathrm{b}\). Let \(\mathrm{B}=\left\{\mathrm{f}_{\mathrm{ab}} \mid \mathrm{a}, \mathrm{b} \in \mathbb{R}, \mathrm{a} \neq 0\right\}\). Then \(B\) is a group with respect to the composition of functions. Check whether or not \(A=\left\{f_{\mathrm{ab}} \mid \mathrm{a} \in \mathbb{Q}^{+}, \mathrm{b} \in \mathbb{R}\right\}\) is a normal subgroup of \(B\).

b) Explicitly give the elements and structure of the group \(S_{n} / A_{n}, n \geq 5\).

c) Let \(G\) be a group of order 56. What are all its Sylow p-subgroups? Show that \(G\) is not simple, i.e., \(\mathrm{G}\) must have a proper normal non-trivial subgroup.

5. a) Find a group \(G\), and a homomorphism \(\phi\) of \(G\), so that \(\phi(G) \simeq S_{3}\) and \(\operatorname{Ker} \phi \simeq \mathrm{A}_{4}\). Is \(G\) abelian? Give reasons for your answer.

b) Let \(\mathrm{G}\) be a group such that Aut \(\mathrm{G}\) is cyclic. Prove that \(\mathrm{G}\) is abelian.

6. a) Check whether \(I=\left\{\left[\begin{array}{cc}m & 0 \\ n & 0\end{array}\right] \mid m, n \in \mathbb{Z}\right\}\) is a subring of the ring \(\mathbb{M}_{2}(\mathbb{Z})\) or not. If it is, check whether or not it is an ideal of the ring also. If I is not a subring of the ring, then provide a subring of the ring.

b) Prove that \(\frac{\mathbb{R}[x]}{\left\langle x^{2}+1\right\rangle} \simeq \mathbb{C}\) as rings.

c) Find all the units of \(\mathbb{Z}_{12}\).

7. a) Let \(R\) be a commutative ring with unity and \(r \in R\). Prove that \(\frac{R[x]}{\langle x-r\rangle} \simeq R\) using the Fundamental Theorem of Homomorphism.

Hence show that \(\frac{R[x, y]}{\langle y-r\rangle} \simeq R[x]\)

b) Let \(\mathrm{D}=\{\mathrm{f}(\mathrm{x}, \mathrm{y})+\mathrm{g}(\mathrm{x}, \mathrm{y})\) i \(\mid \mathrm{f}, \mathrm{g} \in \mathbb{Z}[\mathrm{x}, \mathrm{y}]\} \subseteq \mathbb{C}[\mathrm{x}, \mathrm{y}]\). Check whether \(D\) is a UFD or not.

8. a) Let \(R=\mathbb{Z}[\sqrt{2}]\) and \(M=\{a+b \sqrt{2} \in R|5| a\) and \(5 \mid b\}\).

i) Show that \(\mathrm{M}\) is an ideal of \(\mathrm{R}\).

ii) Show that if \(5 / \mathrm{a}\) or \(5 / \mathrm{b}\), then \(5 /\left(\mathrm{a}^{2}+\mathrm{b}^{2}\right)\), for \(a, b \in \mathbb{Z}\).

iii) Hence show that if \(N\) is an ideal of \(R\) properly containing \(M\), then \(N=R\). iv) Show that \(\mathrm{R} / \mathrm{M}\) is a field, and give two distinct non-zero elements of this field.

b) Show that there are infinitely many values of \(\alpha\) for which \(x^{7}+15 x^{2}-30 x+\alpha\) is irreducible in \(\mathbb{Q}[\mathrm{x}]\).