301.5G Transpositions and the Alternating Group

Подписаться 17 тыс.

Просмотров 7 тыс.

50% 1

Every permutation is a product of transpositions (2-cycles), unique up to the parity -- even/odd -- of the number used. That parity defines the sign of a permutation, and the alternating group consists of all those permutations whose sign is +1 (even number of transpositions).

Опубликовано:

21 июл 2024

Ссылка:

Скачать:

Готовим ссылку...

Добавить в:

Мой плейлист

Посмотреть позже

Комментарии : 10

@ponyboy4416 3 года назад

I like this and am so glad it exists. Sub.

@rosannaa.daitao3764 4 года назад

This is great! Thanks a lot...

@normmacdonaldfan 2 года назад

Absolute GEM

@mateusbalotin7247 Год назад

Hey, aren't you the dude singing that group song hahaha. Thank you for the class!! Exactly what I was looking for. Hope you have an amazing week!!!😁

@MatthewSalomone Год назад

You are correct good sir 😇

@gaaraofddarkness 4 года назад

Hello sir I dont follow the proof of An being a Subgroup of Sn fully... What if there is a transposition which is common to both σ and τ?

@MatthewSalomone 4 года назад

Then there would be a pair (an even number!) of that transposition, and even if στ were rearranged so as to cancel that transposition, the total # of transpositions would still be even for that reason. So the best answer is (though I didn't prove it here): The sign of a permutation is *independent* of which product of transpositions we choose to express it.

@choopa_noopa Год назад

for the subroup test... Say we swap sigma and tao to a and b respectively. Are you not instead supposed to show that not a,b exist in A_n but also ab^(-1) exists in A_n?

@kmn1794 Год назад

A permutation and its inverse have the same parity and so if the alternating group of a symmetric group is all permutations having even parity then it must also include all the inverses.