I Permutations written as product of 2-cycles

[Kaguro]

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.

 

[fresh_42]

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.

 

[Kaguro]

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) ?

 

[fresh_42]

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.

 

[Kaguro]

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.