Abstract Algebra: Permutation Groups Quiz

5 Questions | Total Attempts: 717

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Abstract Algebra: Permutation Groups Quiz - Quiz


Questions and Answers
  • 1. 
    Let Sn be the symmetric group on a finite set with 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).

  • 2. 
    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.

  • 3. 
    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.

  • 4. 
    Which of the following is(are) CORRECT? 1. There exists a permutation α in Sn, such that α(1 2)α-1 = (1 2 3).2. There exists NO permutation α in Sn, such that α(1 2 3)α-1 = (4 5 6).3. If α∈ Sand |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

  • 5. 
    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.

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