S5 Permutations: Can a & b Create Cycle d?

In summary, two elements can create a cycle in an S5 permutation if they are both present in the permutation and not already part of a different cycle. To determine if a and b can create a cycle, check if they meet these conditions. A cycle in an S5 permutation is a sequence of elements that are permuted in a specific order. A and b can be part of multiple cycles in an S5 permutation if they are present multiple times or part of a larger cycle. The maximum number of cycles that can be created in an S5 permutation is 5.
  • #1
bokasabi
4
0

Homework Statement



can cycles a and b create cycle d?
let cycle a = 123
cycle b=12345
cycle d = 12

i.e. can some combination of a and b = d

Homework Equations



only working in S5

The Attempt at a Solution



I have tried various different permutations, 23 over all, things like aba, ab(a^-1), bab, (b^2)a(b^2)
things like this. I am not sure if a solution exists but I have been trying differnt guesses for hours. I don't want to try 120 times to get it.
 
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  • #2
Do you know what odd and even permutations are?
 
  • #3
Good hint!
 

FAQ: S5 Permutations: Can a & b Create Cycle d?

1. Can two elements create a cycle in an S5 permutation?

Yes, two elements can create a cycle in an S5 permutation as long as they are both present in the permutation and are not already part of a different cycle.

2. How do I know if a and b can create a cycle in an S5 permutation?

To determine if a and b can create a cycle in an S5 permutation, you can check if they are both present in the permutation and if they are not already part of a different cycle. If both conditions are met, then a and b can create a cycle.

3. What is a cycle in an S5 permutation?

A cycle in an S5 permutation is a sequence of elements that are permuted in a specific order. For example, if the permutation is (1 3 5 2 4), then the cycle would be (1 3 5 2 4), as each element is permuted to the next one in the cycle.

4. Can a and b be part of multiple cycles in an S5 permutation?

Yes, a and b can be part of multiple cycles in an S5 permutation. This can happen if they are present in the permutation multiple times or if they are part of a larger cycle that contains them.

5. How many cycles can be created in an S5 permutation?

The number of cycles that can be created in an S5 permutation depends on the specific permutation. In general, the maximum number of cycles that can be created in an S5 permutation is 5, as each element can only be part of one cycle.

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