# IGNOU MTE-06 Solved Assignment 2023 | MTE

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

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Details For MTE-06 Solved Assignment

## IGNOU MTE-06 Assignment Question Paper 2023

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 $$\mathbb{Q}[\mathrm{x}]$$.

$$Sin^2\left(\theta \:\right)+Cos^2\left(\theta \right)=1$$

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$$cos\:2\theta =cos^2\theta -sin^2\theta$$

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