Question

For σ ∈ S₈, let o(σ) denote the order of σ. Then max{o(σ) : σ ∈ S₈} is equal to __________.

Answer 1

Correct Answer: 15

To find the maximum order of an element σ in the symmetric group S₈, we need to understand that the order of σ is the least common multiple (LCM) of the lengths of the cycles in its cycle decomposition. To maximize this order, we should use the largest possible cycle lengths whose sum is 8.

Consider breaking down the integer 8 into cycle lengths: 

• One cycle: (8). The order is 8.
• Two cycles: (7,1). The order is LCM(7,1) = 7.
• Two cycles: (6,2). The order is LCM(6,2) = 6.
• Two cycles: (5,3). The order is LCM(5,3) = 15.
• Three cycles: (4,2,2). The order is LCM(4,2,2) = 4.
• Three cycles: (3,3,2). The order is LCM(3,3,2) = 6.
• Four cycles: (2,2,2,2). The order is LCM(2,2,2,2) = 2.

From these calculations, the maximum order possible is 15 from the cycle structure (5,3). Since 15 is both the computed maximum order and within the given range [15,15], it is the correct answer.

Conclusion: max{o(σ) : σ ∈ S₈} = 15.