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.
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?
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.
I think you're thinking of the *one-step* subgroup test. There's also the *two-step* subgroup test and the *finite* subgroup test. ua-cam.com/video/9kQw4tY-z1I/v-deo.htmlfeature=shared
I think Professor Salomone is correct that (26745) = (26)(27)(24)(25). At 3:37 he says he's using the formalism where he's reading the transpositions from left to right, and working it out on paper that's what I get too.
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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?
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.
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?
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.
I think you're thinking of the *one-step* subgroup test. There's also the *two-step* subgroup test and the *finite* subgroup test.
ua-cam.com/video/9kQw4tY-z1I/v-deo.htmlfeature=shared
Q: What's green and sings?
A: Elvis Parsley!
(I still remember that was posted somewhere in my elementary school)
Absolute GEM
This is great! Thanks a lot...
I like this and am so glad it exists. Sub.
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!!!😁
You are correct good sir 😇
Yeah that song was awesome, and it was fun to just stumble across that while going through his group theory playlist 😆
I’m sorry but I think
(26745)=(25)(24)(27)(26).
I think you have it backwards.
I think Professor Salomone is correct that (26745) = (26)(27)(24)(25). At 3:37 he says he's using the formalism where he's reading the transpositions from left to right, and working it out on paper that's what I get too.