Two questions about cycles (algebra)

In summary, the conversation discusses the application of the theorem on cycles and how it differs from the product of individual symbols in the cycle. It also explains how the numbers in the tables on the second page were computed, using the example of the permutation group S_4.
  • #1
Artusartos
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I have two questions:

1) For the example on the second page, I don't understand why they say [tex]\alpha\gamma\alpha^{-1} = (\alpha1 \alpha3)(\alpha2 \alpha4 \alpha7)(\alpha5)(\alpha6)[/tex] instead of [tex]\alpha\gamma\alpha^{-1} = (\alpha1\alpha^{-1} \alpha3\alpha^{-1})(\alpha2\alpha^{-1} \alpha4\alpha^{-1} \alpha7\alpha^{-1})(\alpha5\alpha^{-1})(\alpha6\alpha^{-1})[/tex].

2) For the tables at the top of the 2nd page, I don't know how they computed those numbers...

Thanks in advance
 

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  • #2
Artusartos said:
I have two questions:

1) For the example on the second page, I don't understand why they say [tex]\alpha\gamma\alpha^{-1} = (\alpha1 \alpha3)(\alpha2 \alpha4 \alpha7)(\alpha5)(\alpha6)[/tex] instead of [tex]\alpha\gamma\alpha^{-1} = (\alpha1\alpha^{-1} \alpha3\alpha^{-1})(\alpha2\alpha^{-1} \alpha4\alpha^{-1} \alpha7\alpha^{-1})(\alpha5\alpha^{-1})(\alpha6\alpha^{-1})[/tex].

They say what is consistent with what the theorem says. The theorem says to "apply [itex] \alpha [/itex]" to the symbols in the cycles.

If [itex] \alpha,\ p,\ q [/itex] are cycles, It is true that [itex] \alpha (\ p \ q) \ \alpha^{-1} =( \alpha \ p \ \alpha^{-1})(\alpha \ q \ \alpha^{-1}) [/itex] but this is not the content of the theorem. A cycle is not the same as the product of the individual symbols in the cycle. The cycle (1,2,3) is not equal to (1)(2)(3).

2) For the tables at the top of the 2nd page, I don't know how they computed those numbers...

For example, In the permutation group [itex] S_4 [/itex], there are 8 different elements of the group that are cycles of length 3. The example (1,2,3) in the table illustrates one of them.
(There are 24 = (4)(3)(2) different permutations that can be formed by taking 3 distinct numbers from the set of numbers {1,2,3,4}. However, each permutation such as (1,2,3) is one of 3 representations of the same cycle. (1,2,3) = (2,3,1) = (3,1,2) So there are 8 = 24/3 distinct cycles of length 3 )
 
  • #3
In a nutshell, notice that alpha gamma alpha inverse takes alpha of 1 to alpha of 3. ;)
 
  • #4
Stephen Tashi said:
They say what is consistent with what the theorem says. The theorem says to "apply [itex] \alpha [/itex]" to the symbols in the cycles.

If [itex] \alpha,\ p,\ q [/itex] are cycles, It is true that [itex] \alpha (\ p \ q) \ \alpha^{-1} =( \alpha \ p \ \alpha^{-1})(\alpha \ q \ \alpha^{-1}) [/itex] but this is not the content of the theorem. A cycle is not the same as the product of the individual symbols in the cycle. The cycle (1,2,3) is not equal to (1)(2)(3).



For example, In the permutation group [itex] S_4 [/itex], there are 8 different elements of the group that are cycles of length 3. The example (1,2,3) in the table illustrates one of them.
(There are 24 = (4)(3)(2) different permutations that can be formed by taking 3 distinct numbers from the set of numbers {1,2,3,4}. However, each permutation such as (1,2,3) is one of 3 representations of the same cycle. (1,2,3) = (2,3,1) = (3,1,2) So there are 8 = 24/3 distinct cycles of length 3 )

Thank you
 

1. What is a cycle in algebra?

A cycle in algebra refers to a repeating pattern or sequence of numbers or variables. It can also be represented graphically as a closed loop.

2. How can I identify cycles in an equation?

To identify cycles in an equation, look for a repeating pattern or sequence of numbers or variables. You can also plot the equation on a graph and look for a closed loop.

3. What is the difference between a cycle and a periodic function?

A cycle is a repeating pattern or sequence, while a periodic function is a function that repeats itself after a specific interval. In other words, a cycle refers to the visual representation of a pattern, while a periodic function refers to the mathematical concept of repetition.

4. Can cycles be applied to real-life situations?

Yes, cycles can be applied to real-life situations. For example, they can be used to represent the cyclical patterns of seasons, stock market trends, or the phases of the moon.

5. How are cycles useful in algebra?

Cycles are useful in algebra as they help to identify patterns and make predictions. They also allow us to solve equations and understand the behavior of functions over time.

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