Permutations - determine order of S_n

In summary, the question is asking for the largest number which is the order of an element in the group S_8, and to write down an element with that order in disjoint cycle notation. The order of an element in a group is the smallest number n such that the element to the power of n equals the identity element. The largest possible order for S_8 is 15, and an example element with that order could be (12345)(678).
  • #1
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Homework Statement

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What is the largest number which is the order of an element of S_8? Write down
an element of that order in disjoint cycle notation.


Homework Equations





The Attempt at a Solution


To start with, I don't understand the wording of the question. When it refers to element, does it imply permutation. If so, then is the question asking that we find the composition of S_8 such that the order is at its largest (wherbey the order is the product of the least common multiple of the cycle lengths)?

For ex,
(12345)(678)
The order is 15
 
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  • #2
Recall that the order of an element of a group is the order of the group generated by that element. Equivalently, if a is an element of some group, then its order is the smallest n such that [itex]a^n=e[/itex], where e is the identity.
 
  • #3
So you're suggesting that the largest order for S_8 is 8?

But can it be 15? Referring back to my example...
 
  • #4
Elements of [tex]S_8[/tex] are permutations, as you mentioned. Each element of [tex]S_8[/tex] has an order, of all possible orders of elements, the question asks for the largest. I think you would be right that it is in fact 15.
 

FAQ: Permutations - determine order of S_n

1. What is the definition of S_n in permutations?

S_n, also known as the symmetric group, is a mathematical group that represents all possible arrangements of a set of n distinct objects.

2. How do you determine the order of S_n?

The order of S_n can be determined by taking the factorial of n, denoted as n!, which represents the number of possible permutations in S_n.

3. Can the order of S_n be calculated for any value of n?

Yes, the order of S_n can be calculated for any positive integer value of n. However, for larger values of n, the calculation may become more complex.

4. How is the order of S_n related to the number of permutations?

The order of S_n is equal to the number of permutations in S_n. This is because each permutation corresponds to a unique element in S_n and vice versa.

5. Can the order of S_n be used to find the number of possible combinations?

No, the order of S_n only represents the number of permutations. To find the number of possible combinations, the order would need to be divided by the number of repetitions, if any, in the set of objects.

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