Solve your IGNOU Doubts

Question:

Which of the following statements are true and which are false? Justify your answer with a short proof or a counter example.

(a) Every subgroup of S3 is normal.

(b) Every abelian group is cyclic.

(d) If a field has characteristic p, p a prime, the field is finite.

(e) If every element in group has finite order, the group is finite.

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Question:

Let R be a ring (not necessarily commutative) and I and J be ideal of R. Show that I $\cap$ J $I+J=\{a+b\mid a\in I,b\in J\}$ and I + J = } are ideals of R.

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Question:

Check whether or not $\left(\bar{3}\right)$ is a maximal ideal in $\mathbb{Z}$ _^9.

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Question:

Show that $(x,5)$ is not a principal ideal in Z[x].

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Question:

$\text{Let}S^1=\{z\in\mathbb{C}^*\mid|z|=1\}\text{and}U_n=\{z\in\mathbb{C}^*\mid z^n=1\}\text{for}n\in\mathbb{N}.\text{Check that}U_n\subseteq S^1.\text{Further,show that}U_n\leq S^1.$

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Question:

Let F be a field and let f(x) ∈ F[x] be irreducible in F[x]. Show that the ideal (f(x)) is a maximal ideal in F[x]. Use this to deduce that Q[x]/(x² + 6x³ +12) is a field.

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Question:

Find the order of each of the elements in U(15). Is U(15) cyclic? Justify your answer.

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Question:

If H and K are normal abelian subgroups of a group, and if H∩K = {e}, show that HK is abelian. Will the result be still true if we remove the condition that H and K are normal? Justify your answer.

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Question:

Find the ged of the polynomials $x^4+3x+2\quad\text{and}\quad x^3+3x^2+5x+3.$

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Question:

$\text{Let}R=\left\{\begin{bmatrix}a&b\\0&c\end{bmatrix}\middle|a,b,c\in\mathbb{R}\right\}.$ Check whether R is a subring of $M_2(\mathbb{R})?$ ? Justify your answer

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Question:

Show that, if G is a finite group and $a\in G,\quad o(a)\mid o(G).$ Further, show that $a^{o(G)}=e\text{for}$ $\text{all}a\in G.$ Deduce the Euler-Fermat theorem $a^{\phi(n)}\equiv 1\pmod{n}\text{for all}a,n\in\mathbb{N},n\geq 2,$ $(a,n)=1.$

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Question:

Calculate the following:

(i) $\quad(\overline{3}x^2+\overline{4}x+\overline{1})+(\overline{3}x^3+\overline{4}x^2+\overline{2}x+\overline{3})\text{in}\mathbb{Z}_5[x].$

(ii) $\quad(\overline{3}x^2+\overline{2}x+\overline{6})\cdot(\overline{3}x^3+\overline{4}x+\overline{5})\text{in}\mathbb{Z}_7[x].$

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Question:

Define an integral domain. Give an example of an integral domain which is not a field.

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Question:

Let R $\text{which}a^2=a\text{for all}a\in R.\text{Show that}a=-a\text{and}R$ is commutative.

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Question:

$\text{Let}R=\mathbb{Z}_{20}.$

(i) Give, with justificaiton, a nilpotent element in R.

(ii) Give, with justification, a zero divisor in 푅 which is not nilpotent.

(iii) What is the order of U(R)?

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Question:

$\text{If}F\text{is a field,show that}U(F[x])=F^*.$

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Question:

$\text{Let}\alpha=(1\;2\;5),\quad\beta=(1\;4\;3\;2)\in S_5.\text{Compute}\sigma=\alpha\cdot\beta^{-1}.\text{Write}\sigma$

as a product of transpositons. What is the signature of $\sigma$ ?

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Question:

$\text{Let}A\text{be a}3\times 4\text{real matrix,}B\text{be a}4\times 2\text{real matrix.}$ and 퐶 be a 2 × 3 real matrix. Which of the following operations are defined?

$(i)\quad CA+B^T\\$

$(ii)\quad AB+C^T$

For those operations that are defined, what is the order of the resulting matrix?

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Question:

$\text{Let}S=\{1,2,3,4\}$ and ∗ be the binary operation defined by $a*b=a.$ Compute the Cayley table fo $(S,*).Is*$ commutative? Is ∗ associative? Justify your answers.