Question

Which of the following statements are true for group of permutations?
A. Every permutation of a finite set can be written as a cycle or a product of disjoint cycles
B. The order of a permutation of a finite set written in a disjoint cycle form is the least common multiple of the lengths of the cycles
C. If \(A_n\) is a group of even permutation of n-symbol (\(n>1\)), then the order of \(A_n\) is n!
D. The pair of disjoint cycles commute

• A. A, B and D only
• B. A, B and C only
• C. A, B, C and D
• D. B, C and D only

Hint:

Memorize the key facts about \(S_n\) and \(A_n\):
Any permutation is a product of disjoint cycles.
Order of a permutation is the LCM of its disjoint cycle lengths.
Disjoint cycles always commute.
\(|S_n| = n!\)
\(|A_n| = n!/2\) for \(n \ge 2\). These four facts answer the vast majority of basic questions about permutation groups.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question tests fundamental theorems and properties of permutation groups, specifically the symmetric group \(S_n\) and the alternating group \(A_n\).

Step 2: Detailed Explanation:
Let's analyze each statement:
A. Every permutation of a finite set can be written as a cycle or a product of disjoint cycles. This is the Fundamental Theorem of Permutation Groups. It states that any permutation has a unique decomposition into disjoint cycles (up to the order of the cycles). This statement is true.
B. The order of a permutation of a finite set written in a disjoint cycle form is the least common multiple of the lengths of the cycles. This is the standard method for calculating the order of a permutation. For example, the order of \((1 2 3)(4 5)\) in \(S_5\) is lcm(3, 2) = 6. This statement is true.
C. If \(A_n\) is a group of even permutation of n-symbol (\(n>1\)), then the order of \(A_n\) is n!. This statement is false. The group of all permutations on n symbols is the symmetric group \(S_n\), and its order is \(|S_n| = n!\). The alternating group \(A_n\) is the subgroup of \(S_n\) containing all even permutations. For \(n \ge 2\), exactly half of the permutations are even and half are odd. Therefore, the order of \(A_n\) is \(|A_n| = \frac{n!}{2}\).
D. The pair of disjoint cycles commute. This is a key property of disjoint cycles. If cycles \(\sigma\) and \(\tau\) are disjoint (meaning they move different sets of elements), then \(\sigma\tau = \tau\sigma\). This is because the action of \(\sigma\) does not affect the elements moved by \(\tau\), and vice versa. This statement is true.
Step 3: Final Answer:
Statements A, B, and D are true, while C is false. Therefore, the correct option is "A, B and D only".