Let R be a commutative ring with unity and re R. Prove that  using

the Fundamental Theorem of Homomorphism.

Hence show that .

Find all the units of Z₁₂.

Prove that   as rings.

Check whether  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.

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

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.

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.

Explicitly give the elements and structure of the group S_n/A_n, n ≥5. 

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. 

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.

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

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

i) (C,.),       ii)  (C, +)

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

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.

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.

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.

समीकरण 12? ² − 2? ² + ? 2 = 2?y को रूपांतरित कीजिए, यदि मूलबिंदु को स्थिर रखा जाए और अक्षों को इस प्रकार घुमाया जाए कि नए अक्षों के दिक-अनुपात 1,-3,0;3,1,0,0,0,1 हों।

जांच कीजिए कि निम्नलिखित शांकवजों में से कौनसे शांकवज केंद्रीय हैं और कौनसे अकेंद्रीय हैं। यह भी पता कीजिए कि जो केंद्रीय शांकवज हैं उनमें से किनके केंद्र मूलबिंदु

पर है।

(i) ? ² + ? ² + ? ² + 4? + 3? − ? = 0

(ii) 2? ² − ? ² − ? ² + ?? + ?? − ?? = 1

(iii) ? ² + ?² − ? ² − 2?? − 3?? − 6?? + ? − 2? + 5? + 4 = 0

दिखाइए कि शांकवज  केंद्रीय है। अतः इसका केंद्र निकालिए।

निर्देशांक अक्षों की दिशाओं को परिवर्तित किए बिना मूलबिंदु को (1,2,0) पर स्थानांतरित करके समीकरण  को रूपांतरित कीजिए। यह नया समीकरण ? ² + 2? ² − 6? ² − 2? − 8? + 3 = 0 क्या निरुपित करता है? इसका स्थूल आरेख बनाइए।