Solve your IGNOU Doubts

Question:

Let R be a commutative ring with unity and r ∈ R . Prove that $\frac{R[x]}{\left \langle x-r \right \rangle}\simeq R$ using the Fundamental Theorem of Homomorphism. Hence show that $\frac{R[x,y]}{\left \langle x-r \right \rangle}\simeq R[x].$

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Question:

Find all the units of $\mathbb{Z}_{12}.$

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Question:

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

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Question:

Check whether $\left \{ \begin{bmatrix} m &0 \\ n & 0 \end{bmatrix}\mid m,n\in \mathbb{Z} \right \}$ 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.

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Question:

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

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Question:

Find a group G , and a homomorphism « of G , so that $\O (G)\simeq S_3$ and Ker $\simeq A_4.$ Is G abelian? Give reasons for your answer.

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Question:

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.

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Question:

Explicitly give the elements and structure of the group $S_n/A_n,n\geq 5.$

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Question:

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

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Question:

Let G be a group of order n ≥ 2 , with only two subgroups $-\left \{ e \right \}$ 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.

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Question:

Let $\left ( G,. \right )$ be a finite abelian group and $m\in \mathbb{N}.$ Prove that $S=\left \{ g\in G\mid (o(g),m)=1 \right \}\leq G.$

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Question:

Check whether or not A $=\left \{ z\in \mathbb{C}^{*}\mid \mid z\mid \in \mathbb{Q}\right \}$ is a subgroup of

i) $\left ( \mathbb{C}^{*},. \right ),$ ii) $\left ( \mathbb{C},+ \right )$

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Question:

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

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Question:

Consider the set $X=\mathbb{R}\setminus \left \{ -1 \right \}.$ Define $*$ on $X$ by $x_1*x_2=x_1+x_2+x_1x_2\forall x_1,x_2\in X.$

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

ii) Prove that x ∗ x ∗ x ∗K∗ x (n times) = (1+ x) −1 ∀ n∈N n and x ∈X .

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Question:

State whether the following statement are true or false. Justify your answer with the help of a short proof or a counter-example.

i) The initial value problem

$\frac{dy}{dx}=x^2+y^2,y(0)0$

has a unique solution in some interval of the form - $h<x<h.$

ii) The orthogonal trajectories of all the parabolas with vertices at the origin and foci on the

$x-axis$ is $x^2+2y^2=c^.$

iii) The normal form of the differential equation

$y^{''}-4xy'+(4x^2-1)y=-3ex^2 sin 2x is \frac{d^2v}{dx} +v=-3sin2x,$

where $v=ye^{-x2}.$

iv The solution of the pde $\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}=z^2$ is $z=[y+f(x-y)].$

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Question:

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