IGNOU MTE 6 SOLVED ASSIGNMENT

MTE 6: Abstract Algebra

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Title Name	IGNOU MTE 6 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSC
Course Name	Bachelor in Science
Subject Code	MTE 6
Subject Name	Abstract Algebra
Year	2026
Session	-
Language	English Medium
Assignment Code	MTE 6/Assignment-1/2026
Product Description	Assignment of BSC (Bachelor in Science) 2026. Latest MTE 06 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU BEGC-131 (BAG) 2025-26 Assignment is for January 2026 Session: 30th September, 2026 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
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MTE 6 2025 - English

Assignment

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) (n) = n-1∀n ∈ N, where o is the Euler-phi function.

ii) If G₁ and G₂ are groups, and f: G₁→ G₂ is a group homomorphism, then 2 o(G₁) = 0(G2).

If G is an abelian group, then G is cyclic.

iv) If G is a group and HAG, then | G: H=2.

v) Every element of S has order at most n.

vi) If R is a ring and I is an ideal of R, then xr = rx ∀ x ∈ I and re R.

vii) If σε S_n(n ≥3) is a product of an even number of disjoint cycles, then sign (σ)=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 R form a ring with respect to point- wise addition and multiplication.

2. a) Define a relation R on Z, by R = {(n, n+3k)|k∈ Z}.

Check whether R is an equivalence relation or not. If it is, find all the distinct equivalence classes. If R is not an equivalence relation, define an equivalence relation on Z.

b) Consider the set X = R {-1}. Define * on X by

X₁ * X2 = X + X2 + X₁X2 X1, X2 Ε Χ.

i) Check whether (X, *) is a group or not.

ii) Prove that x * x*x*...*x (n times) = (1+x)" -1∀ n∈ N and x ∈ X.

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

3. a) Check whether or not A = {z∈ C^*I IzI∈Q} is a subgroup of

i) (C,.), ii) (C, +)

b) Let (G,.) be a finite abelian group and me N. Prove that S={g∈Gl(o(g), m) = 1} <G.

c) Let G be a group of order n ≥ 2, with only two subgroups - {e} and itself. Find a minimal generating set for G. Also, find out whether n is a prime or a composite number, or can be either.

a) Consider the map f_ab: R→R: f_ab(x)=ax+b. Let B = {f_ab, Ia, b ∈ R, a≠0}. Then B is a group with respect to the composition of functions. Check whether or not A={f_ab I a ∈ Q⁺, b∈ R} is a normal subgroup of B.

b) Explicitly give the elements and structure of the group S_n/A_n, n ≥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., G must have a proper normal non-trivial subgroup.

5. a) Find a group G, and a homomorphism Ø of G, so that Ø (G)=S3 and Ker Ø = A4. Is G abelian? Give reasons for your answer.

b) Let G be a group such that Aut G is cyclic. Prove that G is abelian.

6. a) Check whether $I=left{egin{bmatrix}m&0\n&0end{bmatrix}middle|m,ninmathbb{Z} ight}$ is a subring of the ring M₂(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 $frac{mathbb{R}[x]}{langle x^2+1 angle}simeqmathbb{C}$ as rings.

c) Find all the units of Z₁₂.

7. a) Let R be a commutative ring with unity and re R. Prove that $frac{R[x]}{langle x-r angle}simeq R$ using

the Fundamental Theorem of Homomorphism.

Hence show that $frac{R[x,y]}{langle y-r angle}simeq R[x]$ .

b) Let $D={f(x,y)+g(x,y)imid f,ginmathbb{Z}[x,y]}subseteqmathbb{C}[x,y]$ . Check whether D is a UFD or not.

8. a) Let R = [√2] and M = (a+b√2∈ R 51a and 51b).

i) Show that M is an ideal of R.

ii) Show that if 5 Ja or 5]b, then 5] (a²+b²), for a, b∈ Z..

iii) Hence show that if N is an ideal of R properly containing M, then N = R.

iv) Show that R/M is a field, and give two distinct non-zero elements of this field.

b) Show that there are infinitely many values of a for which x² +15x²-30x + a is irreducible in Q[x].

MTE 6 2025 - Hindi

सत्रीय कार्य

(इसे चारों खंडों को पढ़ने के बाद कीजिए।)

iB Øe di, e VhbZ-06

I=h dk ZdM:, e VhbZ 6Vh, e, /2026

vi/ldre मान: 100

1. निम्नलिखित में से कौन-से कथन सत्य है? अपने उत्तरों की पुष्टि कीजिए। (इसका अर्थ है कि यदि आप सोचते हैं कि कोई कथन असत्य है, तो एक संक्षिप्त उपपत्ति या एक उदाहरण ऐसा दीजिए जो उसे असल्य दर्शाए। यदि यह एक सत्य कथन है, तो तो ऐसा कहने। ऐसा कहने के लिए एक संक्षिप्त उपपत्ति दीजिए।)

(i) $(n)=n-1,quadforall ninmathbb{N}_{+}$ जहाँ φ के ऑयलर-फाई फलन है।

ii) यदि G₁, और G₂, समूह हैं, तथा f:G₁, G_², एक समूह समाकारिता है, तो ० (G₁)=o(G₂) होगा।

iii) यदि G एक आबेली समूह है, तो G चक्रीय होगा।

iv) यदि G एक समूह है तथा $Hunderline{A}G,$ तो | G:H|= 2 होता है।

v) S_n के प्रत्येक अवयव की कोटि ज़्यादा से ज़्यादा होती है।

vi) यदि R एक वलय है तथा R की एक गुणजावली 1 है, तो $xr=rxquadforall xin I$ और $repsilon R.$

vii) यदि $sigmain S_nquad(ngeq 3)$ सम संख्या में असंयुक्त चक्रों का गुणनफल है, तो sign(Q) = 1 होता है।

viii) यदि किसी वलय का एक मात्रक है, तो उसका केवल एक ही मात्रक होता है।

ix) परिमित क्षेत्र का अभिलक्षणिक शून्य होता है।

x) [0, 1] से R तक असंतत फलनों का समुच्चय बिंदुशः योग और गुणन के सापेक्ष एक वलय होता है।

2. (क) $R={(n,n+3k)mid kinmathbb{Z}}$ द्वारा $Z$ पर एक संबंध R परिभाषित कीजिए। जाँच कीजिए कि R एक तुल्यता संबंध है या नहीं। यदि है, तो सभी अलग-अलग तुल्यता वर्ग ज्ञात कीजिए। यदि R. एक तुल्यता संबंध नहीं है, तो $Z$ पर एक तुल्यता संबंध परिभाषित कीजिए।

(ख) समुच्चय $X=mathbb{R}setminus{-1}$ पर विचार कीजिए। X पर ∗ को

$X_1*X_2=X_1+X_2+X_1X_2,quadforall X_1,X_2in X$ द्वारा परिभाषित कीजिए।

i) जाँच कीजिए कि (X,*) एक समूह है या नहीं।

ii) सिद्ध कीजिए कि x*x*x*...*x (n बार) = $(1+x)^n-1quadforall ninmathbb{N},xin X.quad$

(ग) पुष्टिकरण के साथ, किन्सी अक्रमविनिमेय समूह के एक क्रमाविनिमेय उपसमूह का उदाहरण दीजिए।

3.(क) जाँच कीजिए कि $A={zinmathbb{C}^*mid|z|inmathbb{Q}}$ निम्नलिखित का एक उपसमूह है या नहीं:

i) $(mathbb{C}^*,cdot)$ ii) $(mathbb{C},+)$

(ख) मान लीजिए कि (G,.) एक परिमित आबेली समूह है तथा $mN.$ सिद्ध कीजिए कि $S={gin Gmid( ext{o}(g),m)=1}leq G$

(ग) मान लीजिए कि G कोटि $nleq 2$ वाला एक समूह है, जिसके केवल दो उपसमूह हैं $-left{e ight}$ और स्वयं G. G के लिए एक अल्पिष्ठ जनक समुच्चय ज्ञात कीजिए। साथ ही, यह भी ज्ञात कीजिए कि क्या । एक अभाज्य संख्या है, या भाज्य संख्या है, या दोनों में से कोई भी हो सकती है।

4. (क) फलन $f_{ab}:mathbb{R} omathbb{R}:f_{ab}(x)=ax+b$ पर विचार कीजिए। मान लीजिए कि $B={f_{ab}mid a,binmathbb{R},a e 0}$ तब फलन के संजन के सापेक्षा B एक समूह है। जाँच कीजिए कि $A={f_{ab}mid ainmathbb{Q}^+,binmathbb{R}},B$ उपसमूह है या नहीं।

(ख) $S_n/A_n,quad ngeq 5$ के और इसकी संरचनाको स्पष्ट शम में दीजिए।

(ग) मान लीजिए कि G कोटि 56 का एक समूह है। इसके सभी सीलो p-उपसमूह क्या होंगे? दर्शाइए कि G सरल नहीं है, अर्थात् G का एक उचित प्रसामान्य अतुच्छ उपसमूह अवश्य ही होना चाहिए।

5. (क) एक ऐसा समूह G तथा G की एक ऐसी समाकारिता Ø ज्ञात कीजिए, जिससे $phi(G)simeq S_3$ कि हो तथा $ext{Ker}phicong A_4$ हो। क्या G आबेली होगा? अपने उत्तर के लिए कारण दीजिए।

(ख) मान लीजिए कि G एक ऐसा समूह है कि AutG चक्रीय है। सिद्ध कीजिए कि G आबेली होगा।

6. क) मान लीजिए कि $I=left{egin{bmatrix}m&0\n&0end{bmatrix}middle|m,ninmathbb{Z} ight}$ वलय M₂ (Z) का एक उपवलय है या नहीं।

यदि है, तो जाँच कीजिए कि वह इस वलय की एक गुणजावली भी है या नहीं। यदि I इस वलय का एक उपवलय नहीं है, तो इस वलय का एक उपवलय दीजिए।

ख) सिद्ध कीजिए कि वलयों के रूप में, $frac{mathbb{R}[x]}{langle x^2+1 angle}congmathbb{C}$

ग) $Z_{12}$ के सभी मात्रकों को ज्ञात कीजिए।

7.क मान लीजिए कि R एक क्रमविनिमेय तत्समकी वलय है और $repsilon R$ . समाकारिता के मूल प्रमेय का प्रयोग करते हुए, सिद्ध कीजिए कि $frac{R[x]}{langle x-r angle}cong R$ इस तरह, दर्शाइए कि $frac{R[x,y]}{langle y-r angle}cong R[x]$

ख) मान लीजिए कि $D={f(x,y)+g(x,y)imid f,ginmathbb{Z}[x,y]}subseteqmathbb{C}[x,y]$ जाँच कीजिए कि D एक UFD है या नहीं। (4)

8.क) मान लीजिए कि $R=Zleft{sqrt[]{2} ight}$ तथा $M={a+bsqrt{2}in Rmid 5mid a}$ और $5mid a}$

i) दर्शाइए कि M, R की एक गुणजावली है।

ii) दर्शाइए कि यदि 5/ a या 5/b, तो 5/ (a²+b²), जहां a, $bvarepsilon z.$

अत, दर्शाइए कि यदि N. R की एक ऐसी गुणजावली है जिसका M एक उचित उपसमुच्चय है, तो N = R होगा।

iv) दर्शाइए कि $Rsetminus M$ एक क्षेत्र है, तथा इस क्षेत्र के दो अलग-अलग शून्येतर अवयव दीजिए।

ख) दर्शाइए कि के अंततः अनेक मान हैं जिनके लिए $ext{Q}[x] ext{}x^7+15x^2-30x$ अखंडनीय है।

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