Abstract Algebra Answers

List two distinct cosets of < r > in , D₁₀ where r is a reflection in . D1

Let τ be a fixed odd permutation in . S₁₀ Show that every odd permutation in S₁₀ is

a product of τ and some permutation in A₁₀

Using Cayley’s theorem, find the permutation group to which a cyclic group of 

order 12 is isomorphic

Prove that a cyclic group with only one generator can have at most 2 elements.

Let G be an infinite group such that for any non-trivial subgroup H of 

G, G : H < ∞. Then prove that 

 i) H ≤ G ⇒ H = {e} or H is infinite; 

 ii) If g ∈G, g ≠ e, then o(g) is infinite.

Prove that every non-trivial subgroup of a cyclic group has finite index. Hence 

prove that (Q, +) is not cyclic.

Which of the following statements are true? Give reasons for your answers.

 i) If a group G is isomorphic to one of its proper subgroups, then G = Z.

 ii) If x and y are elements of a non-abelian group (G, ∗) such that x ∗ y = y ∗ x, then x = e or y = e, where e is the identity of G with respect to . ∗

 iii) There exists a unique non-abelian group of prime order. 

 iv) If (a, b)∈A× A, where A is a group, then o((a, b)) = o(a)o(b).

 v) If H and K are normal subgroups of a group G, then hk = kh ∀ h ∈H, k ∈K.

Solve the following LPP by the two-phase simplex method.

 Max Z = x₁ + x₂ − x₃

 Subject to 4x₁ + x₂ + x₃ = 4

Which of the following statements are true? Give a short proof or a counter example in 

support of your answer.

 (i) For any two square matrices A and B, AB = BA.

 (ii) If the following table is obtained in the intermediate stage while solving an LPP by  the Simplex method, then the LPP has an unbounded solution: 

____________________________

-1 -2 0 0 0

____________________________

1 x₁ 1 2 -1 0 1

0 x₄ 0 3 -1 1 2

____________________________

0 4 -1 0 1

____________________________

(iii) The number of basic variables in a feasible solution of a transportation problem with  m sources and n destinations is mn.

iv) An optimal assignment of the assignment problem with cost matrix C is also an optimal assignment of the assignment problem with cost matrix C^t

 (v) (1,2) is an optimal solution to the following LPP: 

 Max Z = 2x₁ + 4x₂ subject to 

 x₁ + 2x₂ ≤ 5

 x₁ + x₂ ≤ 4

 x₁, x₂ ≥ 0

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) If A and B are two sets such that A ⊆ B, then A × B = B.

ii) If S is the set of people on the rolls of IGNOU in 2016 and T is the set of real

numbers lying between 2.5 and 2.55, then SUT is an infinite set.

he set {x∈Z| x ≡1(mod30)}is a group with respect to multiplication(mod30).

iv) If G is a group with an abelian quotient group G/N, then N is abelian.

v) There is a group homomorphism f with Ker f ≅ R and Imf ≅{0}.