Permutations written as product of 2-cycles

In summary, the author of Gallian's book on Group Theory provides an example in the chapter on permutation groups, showing that (12345) can be written as (15)(14)(13)(12) and also as (54)(52)(21)(25)(23)(13). This demonstrates that the decomposition of a permutation is not unique. The author also mentions that only the sign (odd/even) is an invariant. The conversation then delves into discussing the lack of a pattern in the decomposition and how the author may have arrived at the simplification of (12345). It is suggested that the author may have taken a representation with 4 transpositions and conjugated one of them. However, the concept of conjugation is
  • #1
Kaguro
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TL;DR Summary
The decomposition of an n-cycle into 2-cycles can be done in various ways. I don't understand how to think of other ways.
I'm trying to learn Group Theory from Gallian's book. When I reached the chapter for permutation groups, the author gives an example that we can write (12345) as (15)(14)(13)(12). I immediately recognized that this should always work (I proved it later.)

Then author says we can write :
(12345) = (54)(52)(21)(25)(23)(13)

I checked, yes this works. But how did the author get such a horrific looking way to write the permutation? I don't see any pattern here.
 
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  • #2
There is no pattern. It is only meant to demonstrate that such a decomposition isn't unique. E.g. ##(52)(21)(25)=(15)##. Only the sign (odd/even) is an invariant.
 
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  • #3
:oops:😐😶
No pattern?

Well, I can see why (52)(21)(25) = (15)
But I can't see why (15) = (52)(21)(25)

I mean, did the author just try a lot of products of six 2-cycles and ended up with one which simplifies to (12345) ?
 
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  • #4
Kaguro said:
:oops:😐😶
No pattern?

Well, I can see why (52)(21)(25) = (15)
But I can't see why (15) = (52)(21)(25)

I mean, did the author just try a lot of products of six 2-cycles and ended up with one which simplifies to (12345) ?
I guess he took a representation with 4 transpositions and simply conjugated one of them.
 
  • #5
fresh_42 said:
I guess he took a representation with 4 transpositions and simply conjugated one of them.
Uhhh...
I have no idea about conjugation yet.
I'll read this again in a few days.

Thanks.
 

1. What is a permutation written as a product of 2-cycles?

A permutation written as a product of 2-cycles is a way of expressing a permutation as a series of 2-cycle transpositions. This means that each 2-cycle represents swapping two elements in the permutation. For example, the permutation (1 2 3 4) can be written as (1 2)(2 3)(3 4).

2. How is a permutation written as a product of 2-cycles useful?

Writing a permutation as a product of 2-cycles can be useful in understanding the structure of the permutation. It can also be helpful in solving certain mathematical problems, such as determining the order of a permutation or finding the inverse of a permutation.

3. Can any permutation be written as a product of 2-cycles?

Yes, any permutation can be written as a product of 2-cycles. This is known as the 2-cycle decomposition theorem, which states that any permutation can be expressed as a series of 2-cycle transpositions.

4. How many 2-cycles are needed to express a permutation?

The number of 2-cycles needed to express a permutation is equal to the number of inversions in the permutation. An inversion occurs when two elements in a permutation are in reverse order compared to their original position. For example, the permutation (1 3 2) has one inversion and can be written as (1 3)(2 3), using two 2-cycles.

5. Can a permutation be written as a product of 2-cycles in multiple ways?

Yes, a permutation can be written as a product of 2-cycles in multiple ways. This is because the order in which the 2-cycles are written does not affect the resulting permutation. Additionally, some permutations can be written as a product of 2-cycles in different lengths, such as (1 2)(2 3)(3 4) or (1 2 3)(3 4).

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