Algebra Test - T 1

Rsquared Mathematics

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Time: 45:00

Topic: Algebra

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The Quiz contains a total of 10 Questions.
Part B(Single Correct) contains 5 questions, each of 3 marks and -0.75 marks for incorrect answers.
Part C(MSQ) contains 5 questions, each with 4.75 marks and no negative marking. The maximum score for the Test is 38.75. Best of Luck.

Part - B.

1. Which of the following is not true?

(a.) Group of Automorphisms of a non-abelian group is non-cyclic.
(b.) Group of Automorphisms of an abelian group is abelian.
(c.) If \(G\) is a non-abelian group of order 6 then the group of automorphisms of \(G\) is isomorphic to \(S_3\).
(d.) If \(\operatorname{Inn}(G)\) is non-trivial then \(G\) must be non-abelian.

Exp: \(G= K_4\)

2. Number of 5 Sylow subgroups of the permutation group \(S_7\) is

(a.) 6
(b.) 16
(c.) 36
(d.) 126

Exp: Order of 5-SSG = 5 and number of elements of order 5 in \(S_7\) is 504.

3. Let \(\beta = (125)(325) \in S_5\), then for what value of \(0 \leq k \leq 5\), we have \(\beta^{k} = \alpha\) such that \(\alpha \& \beta\) have the same order -

(a.) 2
(b.) 3
(c.) 4
(d.) 5

Exp: \(o(\beta) = 2 = o(\beta^{k}) = \frac{2}{\operatorname{gcd}(k, 2)}\), Hence \(k\) must be odd.

4. Let \(G=\{ < a, b > : o(a) = 2, o(b) = 8 \& o(ab) = 2\}\), if \(\operatorname{Z}(G)\) denotes the center of \(G\), then \(\frac{G}{Z(G)}\) is isomorphic to -

(a.) A trivial Group
(b.) \(\mathbb{Z}_{2}\)
(c.) \(\mathbb{Z}_{2} \times \mathbb{Z}_{2}\)
(d.) \(\mathbb{Z}_{4}\)

Exp: \(G \cong D_{n}\)

5. Let \(G=A_{4} \times D_{3}\), then

(a.) 2-SSG is normal in \(G\)
(b.) 3-SSG is normal in \(G\)
(c.) \(G\) has a normal subgroup of order 36.
(d.) None of the above

Exp: \(H = A_{4} \times \mathbb{C}_{3}\) is normal in \(G\).

Part - C

6. Suppose that \(G\) is a group of order 2020. Then choose the correct -

(a.) \(G\) has a unique subgroup of order 101
(b.) \(G\) has a subgroup of order 5
(c.) \(G\) has a subgroup of order 505
(d.) \(G\) can not be a simple group.

Explanation: Use Sylow's theorem to claim that \(G\) has a subgroup of order 5 and a unique subgroup of order 101 (say \(H\)).

7. Which of the following is/are correct -

(a.) Inverse image of a subgroup is a subgroup under Homomorphism
(b.) Inverse image of a normal subgroup is normal under Homomorphism
(c.) Group \(G\) is abelian \(\Leftrightarrow f: G \rightarrow G; \forall x \in G, f(x) = x^{-1}\) is a Homomorphism
(d.) \(\operatorname{Ker}(f)\) is always a normal subgroup of Range(\(f\)), where \(f\) is any Homomorphism from \(G\) to \(G\)

Explanation: Use OST to show (a) and (b). For (d), if \(\operatorname{Ker}(f) = G \Rightarrow R(f) = \{e\}\), so \(\operatorname{Ker}(f)\) need not be a subgroup of Range.

8. Let \(G\) be a group of permutations of the word \(RAJENDRA\) in which \(A\) is fixed, then choose the correct -

(a.) \(G\) has an element of order 5
(b.) \(G\) has an element of order 8
(c.) \(G\) has an element of order 6
(d.) \(G\) has a proper normal subgroup.

Explanation: \(G \cong S_{5}\)

9. Which of the following is/are the possible class equation of a group of order 30 -

(a.) \(1+15+2+2+2+2+2+2+2\)
(b.) \(1+1+1+6+6+15\)
(c.) \(1+1+1+1+1+5+10+10\)
(d.) \(1+1+3+10+15\)

Explanation: Groups of order 30 are \(Z / 30 Z, D_{15}, Z / 5 Z \times S_{3}, Z / 3 Z \times D_{5}\). For (a), \(G = D_{15}\), others cannot be true.

10. Let \(G = \text{Aut}(\frac{2\mathbb{Z} \times 4\mathbb{Z} \times 6\mathbb{Z}}{10\mathbb{Z} \times 24\mathbb{Z} \times 42\mathbb{Z}})\), Then

(a.) \(G\) is abelian
(b.) \(G\) is cyclic
(c.) \(G\) has a normal subgroup of order 24
(d.) \(G\) has a normal subgroup of order 12.

Explanation: \(G \cong \text{Aut}(\mathbb{Z}_{5}) \times \text{Aut}(\mathbb{Z}_{6}) \times \text{Aut}(\mathbb{Z}_{7}) \cong U(5) \times U(6) \times U(7)\)