Abstract Algebra: Permutation Groups Quiz

5 Questions | By ParveenChhikara | Updated: Mar 21, 2022 | Attempts: 950

Questions and Answers

1.

PLEASE READ CAREFULLY.1.There are 5 Multi-SELECT Questions, in which one or MORE options are correct.2. There is no NEGATIVE marking.
2.

Let S_n be the symmetric group on a finite set with n symbols. Then which of the following is(are) TRUE?
- A.
  
  Sn contains a subgroup of order m, where 1 ≤ m ≤ n.
- B.
  
  Sn contains a subgroup of order (n -1)!.
- C.
  
  Sn is non-Abelian for all n ∈ N.
- D.
  
  There exists a cycle of order m > 1, which belong to Z(Sn).
3.

Which of the following is(are) CORRECT?
- A.
  
  There exist three transpositions whose product is the identity e of the symmetric group Sn.
- B.
  
  The order of the product of two transpositions is either 2 or 3.
- C.
  
  If α and β are two disjoint permutations such that αβ = e, then α = β = e.
- D.
  
  Every permutation in Sn can be expressed as a product of disjoint cycles.
4.

Which among the following are TRUE?
- A.
  
  (1 2)(1 3)(1 4)(2 5) is an even permutation.
- B.
  
  An odd permutation is even.
- C.
  
  (1 2 3 . . . n) is an odd permutation, if n is odd.
- D.
  
  There exists a subgroup of Sn with index 2.
5.

Which of the following is(are) CORRECT? 1. There exists a permutation α in S_n,such that α(1 2)α^-1 = (1 2 3).2. There exists NO permutation α in S_n, such that α(1 2 3)α^-1 = (4 5 6).3. If α∈ S_nand |α| = m, then for any permutation we have | βαβ^-1| = m.4. A cycle which is conjugate with (1 2 3 . . . n) is an n-cycle.
- A.
  
  Option1
- B.
  
  Option2
- C.
  
  Option3
- D.
  
  Option4
6.

Which of the following is(are) correct?
- A.
  
  A subgroup H in S4, generated by (1 2 3) and (1 2) is of order 6.
- B.
  
  The largest possible order of elements of the alternating group A5 is 10.
- C.
  
  Each element in the alternating group A4 can be written as a product of 3-cycles,
- D.
  
  There are exactly n!/2 odd permutations in Sn.