Powers of cycles? Orders of elements?

[calvino]

When calculating powers of cycles, are there any easy steps to use in doing so? I mean...is there a typical relationship when calculating (123)^n?

Also, is it safe to say that a ceratin n-cycle should have order n or n-1?

I guess any explanation of orders for group elements would be appreciated. For now I will look some stuff up on mathworld, etc. Thanks for your help.

 

[mathwonk]

isn't it sort of obvious that any cycle of length n has order n?hence raising (123) to the nth power is the same as raising it to the least residue of n mod 3?

 

[tacman]

The powers of cycles are important in understanding the structure of groups and their elements. The power of a cycle is simply the number of times the cycle needs to be applied to an element to return to its original position. For example, the cycle (123) applied to the element 1 would result in 2, then 3, and finally back to 1, so the power of this cycle is 3.

In general, the power of a cycle (a1 a2 ... ak) is equal to the least common multiple of the integers 1, 2, ..., k. This means that the power of the cycle is the smallest number that is divisible by all the integers from 1 to k. So, for the cycle (123), the power is 3, since the least common multiple of 1, 2, and 3 is 3.

When calculating powers of cycles, there are some easy steps to follow. Firstly, you can use the fact that the power of a cycle is equal to its length. So, for the cycle (123), the power is 3. This means that (123)^n will have a power of 3n.

Another useful relationship is that the power of a cycle is equal to the order of the element. The order of an element is the smallest positive integer n such that the element raised to the power of n is equal to the identity element. So, for the cycle (123), the order is 3, since (123)^3 = (1)(2)(3) = (123).

It is not always safe to say that an n-cycle should have an order of n or n-1. This is because the order of an element can also depend on the group it belongs to. For example, in the group of permutations of four elements, the element (123) has an order of 3, but in the group of permutations of five elements, the same element has an order of 5.

In general, the order of an element can be any positive integer that divides the order of the group. This means that the order of an element can be smaller or larger than the length of the cycle it represents.

In conclusion, understanding the powers and orders of cycles is important in understanding the structure of groups and their elements. There are some useful relationships that can be used to calculate powers of cycles, but it is not always safe to assume that the order