Solve your IGNOU Doubts

Question:

Let G be a group such that $\text{Aut } G$ is cyclic. Prove that G is abelian.

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Question:

Find a group G, and a homomorphism $\phi$ of G, so that $\phi(G) \simeq S_3$ and $\text{Ker } \phi \simeq A_4$ . 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 $B = \{f_{ab} \mid a, b \in \mathbb{R}, a \neq 0\}$ . Then B is a group with respect to the composition of functions. Check whether or not $A = \{f_{ab} \mid a \in \mathbb{Q}^+, b \in \mathbb{R}\}$ is a normal subgroup of B.

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Question:

Let G be a group of order $n \geq 2$ , 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.

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Question:

Let $(G, \cdot)$ be a finite abelian group and $m \in \mathbb{N}$ . Prove that

$S = \{g \in G \mid (o(g), m) = 1\} \leq G$ .

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Question:

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

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

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Question:

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

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Question:

Prove that $x x \dots x \,\, (\text{n times}) = (1 + x)^n - 1 \,\, \forall \,\, n \in \mathbb{N}$ and $x \in X$ .

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Question:

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

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Question:

Consider the set $X = \mathbb{R} \setminus \{-1\}$ . Define * onX by

$x_1 x_2 = x_1 + x_2 + x_1x_2 \,\, \forall \,\, x_1, x_2 \in X$ .

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Question:

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

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 $\mathbb{Z}$ .

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Question:

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.) (20)
i) $\phi(n) = n - 1 \,\, \forall n \in \mathbb{N}$ , where $\phi$ is the Euler-phi function.
ii) If G₁ and G₂ are groups, and $f : G_1 \rightarrow G_2$ is a group homomorphism, then $o(G_1) = o(G_2)$ .
iii) If G is an abelian group, then G is cyclic.
iv) If G is a group and $H \trianglelefteq G$ , then $|G : H| = 2$ .
v) Every element of S_n has order at most n.
vi) If R is a ring and I is an ideal of R, then $xr = rx \,\, \forall x \in I$ and $r \in R$ .
vii) If $\sigma \in S_n \,\, (n \ge 3)$ is a product of an even number of disjoint cycles, then $\text{sign }(\sigma) = 1$ .
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 $\mathbb{R}$ form a ring with respect to point-wise addition and multiplication.

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Question:

जाँच कीजिए कि $S, \mathbf{R}$ पर रैखिकता: स्वतंत्र समुच्चय है। (संकेत: समीकरण

$a_0 \cos x + a_1 \sin x + a_2 x \cos x + a_3 x \sin x$ लीजिए।
$x = 0, \pi, \frac{\pi}{2}, \frac{\pi}{4},$ इत्यादि रखिए और a_i के हल कीजिए)

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Question:

जाँच कीजिए कि $S, \mathbf{R}$ पर रैखिकता: स्वतंत्र समुच्चय है। (संकेत: समीकरण

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Question:

मान लीजिए V ऐसे सभी फलनो का समुच्चय है जो $\mathbf{R}$ में दो बार अवकलनीय हैं और $S = \{\cos x, \sin x, x \cos x, x \sin x\}$

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Question:

मूल बिन्दु से समतल $\mathbf{r} \cdot (6\mathbf{i} + 4\mathbf{j} + 2\sqrt{3}\mathbf{k}) + 2 = 0$ के लंब की दिक्कोज्याएं ज्ञात कीजिए।

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Question:

जाँच कीजिए कि $x \in \mathbf{R} \setminus \{2\}$ के लिए $(fog)(x) = x$ और $x \in \mathbf{R} \setminus \{-1\}$ के लिए $(gof)(x) = x$ ।

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Question:

जाँच कीजिए कि $g(x) = \frac{x-1}{2-x}$ द्वारा दिया गया $g : \mathbf{R} \setminus \{2\} \to \mathbf{R}$ सुपरिभाषित और 1-1 है। इसके आगे जाँच कीजिए कि किसी $x \in \mathbf{R}$ के लिए $g(x) \neq -1$ है।