Abstract Algebra | Transpositions and even and odd permutations.

We define a property of elements of the symmetric group. In particular we show that the decomposition of a permutation into transpositions is invariant with respect to the parity of the number of transpositions.
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[@mirkoschultz9347]

At 3:30 you write (1 2 3) = (1 2) (1 3).
Shouldn't it be (1 2 3) = (1 2) (2 3)? That way it would also match the notation above.
For me your (1 2) (1 3) is equal to (1 3 2).
Or am I missing something here?

[@MdeMatematicas]

He is correct : (123)=(12)(13) and Not (12)(23)

[@rasraster]

@Smeitor that gives (132)

[@wjdan6995]

@@MdeMatematicas can you explain why? I couldn't figure out by myself

[@MdeMatematicas]

@@wjdan6995 you need to compute the image of each element under the permutation (12)(13). The image of 1-->2-->2 and the imagen of 2-->1-->3 and the image of 3-->3-->1. So the final permutation is (123)

[@wjdan6995]

@@MdeMatematicas isn't it supposed to compute from the right to the left?
(1 2) (1 3)
1 -> 3 -> 3
3 -> 1 -> 2
2 -> 2 -> 1
so it is: (1 3 2)
right?

[@music_lyrics-ni7ks]

This is brilliant, we never learned this in class!

[@julianaharmatiuk2662]

Ohhhh ok now I get some things of the previous video! I think their order is inverted maybe you should TRANSPOSE them
Haha get it?? 😏

[@music_lyrics-ni7ks]

ROFL

[@georgets5770]

what did he mean with fix a in 12:21 ? I understand that a must be send from right to left so it will be send to b' but how this holds
(1)(a) = b' ? . Unless τ2' ... τr' are equivalent to identity. I cannot think other way.

[@cosmicchaos9431]

There's an actual mistake in the video, the permutations should be computed from right to left.
For the sake of clarity, if we take some of the equations from 7:00 it's easy to see that we can write the permutiation (1 2 3) as a product of transpositions in different ways, e.g.:
(1 2 3) = (1 3)(1 2)
(1 2 3) = (1 2)(2 3)
(1 2 3) = (2 3)(1 3)
And since we are talking about cycles we can change the writing order of the elements in each individual transposition, although by convention the first element in the transposition should be the smaller one, e.g.:
(1 3)(1 2) = (3 1)(1 2)
(1 2)(2 3) = (2 1)(2 3)
(2 3)(1 3) = (3 2)(3 1)
Hope it helps.

[@srinjoysom1413]

some books like herstein uses the right to left convention but some books like galien uses left to right ....so its not a mistake per se but he should have told which conv he was following