# Abstract Algebra: Permutation Groups Quiz

5 Questions | Total Attempts: 717  Settings  • 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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