The order of a permutation cycle

In summary, the order of a k-cycle (a(1),a(2),...,a(k)) is equal to k, as it is the least common multiple of the lengths of the individual cycles. This is based on the theorem of the order of a permutation set written in disjoint cycle form, as stated by Ruffini-1799. The specific values of a(1), a(2), etc. do not affect the order of the k-cycle.
  • #1
Benzoate
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Homework Statement



what is the order of k-cycle (a(1),a(2),...,a(k))


Homework Equations





The Attempt at a Solution



According to the theorem of the order of a permutation: the order of a permutation set written in disjoint cycle form is the least common multiple of the lengths of the cycles.(Ruffini-1799)

in this case , the length of the k-cycle is k, for all the common multiples of a(1), a(2) ... and a(k) would be k. So the order of cycle k is k right?
 
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  • #2
The length of the cycle is k so the order is k. It doesn't matter what a(1) etc are. If I understand your notation.
 

FAQ: The order of a permutation cycle

What is the definition of "the order of a permutation cycle"?

The order of a permutation cycle is the number of elements in the cycle, or the length of the cycle. It represents the minimum number of times the cycle must be applied to return to the original arrangement of elements.

How do you calculate the order of a permutation cycle?

To calculate the order of a permutation cycle, you simply count the number of elements in the cycle. For example, a cycle (1,2,3) would have an order of 3.

Can a permutation cycle have an order of 1?

Yes, a permutation cycle can have an order of 1 if it consists of only one element and it remains unchanged after being applied once. For example, the cycle (5) would have an order of 1.

What is the relationship between the order of a permutation cycle and the size of the set it is acting on?

The order of a permutation cycle can never be greater than the size of the set it is acting on. In fact, the order of a cycle must always be a factor of the size of the set. For example, a cycle of order 3 cannot act on a set with only 2 elements.

How does the order of a permutation cycle affect its properties?

The order of a permutation cycle can affect its properties in several ways. For example, the order of a cycle determines whether it is an even or odd permutation. Additionally, the order of a cycle can also determine its compatibility with other cycles and its ability to be decomposed into smaller cycles.

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