# IGNOU BMTC-134 Solved Assignment 2023 | B.Sc (G) CBCS

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

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Details For BMTC-134 Solved Assignment

## IGNOU BMTC-134 Assignment Question Paper 2023

Course Code: BMTC-134
Assignment Code: BMTC-134/TMA/2023
Maximum Marks: 100

\section{PART-A (MM: 50 Marks)}

(To be done after studying Blocks 1 and 2.)

1. Which of the following statements are true? Give reasons for your answers.

i) If a group $$G$$ is isomorphic to one of its proper subgroups, then $$G=\mathbb{Z}$$.

ii) If $$\mathrm{x}$$ and $$\mathrm{y}$$ are elements of a non-abelian group $$(\mathrm{G}, *)$$ such that $$\mathrm{x} * \mathrm{y}=\mathrm{y} * \mathrm{x}$$, then $$\mathrm{x}=\mathrm{e}$$ or $$\mathrm{y}=\mathrm{e}$$, where $$\mathrm{e}$$ is the identity of $$\mathrm{G}$$ with respect to $$*$$.

iii) There exists a unique non-abelian group of prime order.

iv) If $$(a, b) \in A \times A$$, where $$A$$ is a group, then $$o((a, b))=o(a) o(b)$$.

v) If $$\mathrm{H}$$ and $$\mathrm{K}$$ are normal subgroups of a group $$\mathrm{G}$$, then $$h k=k h \forall \mathrm{h} \in \mathrm{H}, \mathrm{k} \in \mathrm{K}$$.

2. a) Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that $$(\mathbb{Q},+)$$ is not cyclic.

b) Let $$\mathrm{G}$$ be an infinite group such that for any non-trivial subgroup $$\mathrm{H}$$ of $$G,|G: H|<\infty$$. Then prove that

i) $$\mathrm{H} \leq \mathrm{G} \Rightarrow \mathrm{H}=\{\mathrm{e}\}$$ or $$\mathrm{H}$$ is infinite;

ii) If $$g \in G, g \neq e$$, then $$o(g)$$ is infinite.

c) Prove that a cyclic group with only one generator can have at most 2 elements.

3. a) Using Cayley’s theorem, find the permutation group to which a cyclic group of order 12 is isomorphic.

b) Let $$\tau$$ be a fixed odd permutation in $$\mathrm{S}_{10}$$. Show that every odd permutation in $$\mathrm{S}_{10}$$ is a product of $$\tau$$ and some permutation in $$\mathrm{A}_{10}$$.

c) List two distinct cosets of $$\langle\mathrm{r}\rangle$$ in $$\mathrm{D}_{10}$$, where $$\mathrm{r}$$ is a reflection in $$\mathrm{D}_{10}$$.

d) Give the smallest $$n \in \mathbb{N}$$ for which $$A_{n}$$ is non-abelian. Justify your answer.

4. Use the Fundamental Theorem of Homomorphism for Groups to prove the following theorem, which is called the Zassenhaus (Butterfly) Lemma:

Let $$\mathrm{H}$$ and $$\mathrm{K}$$ be subgroups of a group $$\mathrm{G}$$ and $$\mathrm{H}^{\prime}$$ and $$\mathrm{K}^{\prime}$$ be normal subgroups of $$\mathrm{H}$$ and K, respectively. Then

i) $$\mathrm{H}^{\prime}\left(\mathrm{H} \cap \mathrm{K}^{\prime}\right) \triangleleft \mathrm{H}^{\prime}(\mathrm{H} \cap \mathrm{K})$$

ii) $$\mathrm{K}^{\prime}\left(\mathrm{H}^{\prime} \cap \mathrm{K}\right) \triangleleft \mathrm{K}^{\prime}(\mathrm{H} \cap \mathrm{K})$$ iii) $$\frac{\mathrm{H}^{\prime}(\mathrm{H} \cap \mathrm{K})}{\mathrm{H}^{\prime}\left(\mathrm{H} \cap \mathrm{K}^{\prime}\right)} \simeq \frac{\mathrm{K}^{\prime}(\mathrm{H} \cap \mathrm{K})}{\mathrm{K}^{\prime}\left(\mathrm{H}^{\prime} \cap \mathrm{K}\right)} \simeq \frac{(\mathrm{H} \cap \mathrm{K})}{\left(\mathrm{H}^{\prime} \cap \mathrm{K}\right)\left(\mathrm{H} \cap \mathrm{K}^{\prime}\right)}$$

The situation can be represented by the subgroup diagram below, which explains the name ‘butterfly’.

![](https://cdn.mathpix.com/cropped/2023_03_26_c6fc836875abd15c68e4g-4.jpg?height=813&width=1124&top_left_y=427&top_left_x=500)

\section{PART-B (MM: 30 Marks)}

(Based on Block 3.)

5. Which of the following statements are true, and which are false? Give reasons for your answers.

i) For any ring $$R$$ and $$a, b \in R,(a+b)^{2}=a^{2}+2 a b+b^{2}$$.

ii) Every ring has at least two elements.

iii) If $$\mathrm{R}$$ is a ring with identity and $$\mathrm{I}$$ is an ideal of $$\mathrm{R}$$, then the identity of $$\mathrm{R} / \mathrm{I}$$ is the same as the identity of $$\mathrm{R}$$.

iv) If $$\mathrm{f}: \mathrm{R} \rightarrow \mathrm{S}$$ is a ring homomorphism, then it is a group homomorphism from $$(\mathrm{R},+)$$ to $$(\mathrm{S},+)$$.

v) If $$\mathrm{R}$$ is a ring, then any ring homomorphism from $$\mathrm{R} \times \mathrm{R}$$ into $$\mathrm{R}$$ is surjective.

6. a) For an ideal I of a commutative ring $$\mathrm{R}$$, define $$\sqrt{\mathrm{I}}=\left\{\mathrm{x} \in \mathrm{R} \mid \mathrm{x}^{\mathrm{n}} \in \mathrm{I}\right.$$ for some $$\left.\mathrm{n} \in \mathbb{N}\right\}$$. Show that

i) $$\sqrt{\mathrm{I}}$$ is an ideal of $$\mathrm{R}$$.

ii) $$\mathrm{I} \subseteq \sqrt{\mathrm{I}}$$.

iii) $$\mathrm{I} \neq \sqrt{\mathrm{I}}$$ in some cases.

b) Is $$\frac{R}{\mathrm{I}} \times \frac{\mathrm{R}}{\mathrm{J}} \simeq \frac{\mathrm{R} \times \mathrm{R}}{\mathrm{I} \times \mathrm{J}}$$, for any two ideals $$\mathrm{I}$$ and $$\mathrm{J}$$ of a ring $$\mathrm{R}$$ ? Give reasons for your answer. 7. Let $$\mathrm{S}$$ be a set, $$\mathrm{R}$$ a ring and $$\mathrm{f}$$ be a 1-1 mapping of $$\mathrm{S}$$ onto $$\mathrm{R}$$. Define + and $$\cdot$$ on $$\mathrm{S}$$ by: $$\left.x+y=f^{-1}(f(x))+f(y)\right)$$

$$x \cdot y=f^{-1}(f(x) \cdot f(y))$$

$$\forall \mathrm{x}, \mathrm{y} \in \mathrm{S}$$.

Show that $$(\mathrm{S},+, \cdot)$$ is a ring isomorphic to $$\mathrm{R}$$.

\section{PART-C (MM: 20 Marks)}

(Based on Block 4.)

8. Which of the following statements are true, and which are false? Give reasons for your answers.

i) If $$\mathrm{k}$$ is a field, then so is $$\mathrm{k} \times \mathrm{k}$$.

ii) If $$\mathrm{R}$$ is an integral domain and $$\mathrm{I}$$ is an ideal of $$\mathrm{R}$$, then Char (R)= Char (R/I).

iii) In a domain, every prime ideal is a maximal ideal.

iv) If $$\mathrm{R}$$ is a ring with zero divisors, and $$\mathrm{S}$$ is a subring of $$\mathrm{R}$$, then $$\mathrm{S}$$ has zero divisors.

v) If $$R$$ is a ring and $$f(x) \in R[x]$$ is of degree $$n \in \mathbb{N}$$, then $$f(x)$$ has exactly roots in $$\mathrm{R}$$.

9. a) Find all the units of $$\mathbb{Z}[\sqrt{-7}]$$.

b) Check whether or not $$\mathbb{Q}[x] /<8 x^{3}+6 x^{2}-9 x+24>$$ is a field.

c) Construct a field with 125 elements.

$$\sec ^2 \theta=1+\tan ^2 \theta$$

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$$cos^2\left(\frac{\theta }{2}\right)=\frac{1+cos\:\theta }{2}$$

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