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Prerna Chaudhary posted an Question
November 21, 2021 • 20:24 pm
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30 points
30 points
- IIT JAM
- Mathematics (MA)
Prove .....
For 𝑛 > 2, show that every even permutation in 𝑆௡ is a product of 3-cycles.
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Deepak singh 1 Best Answer
We know every 3-cycle in Sn is in An , because (abc)=(ac)(ab)(abc)=(ac)(ab), and by definition An consists of everything that can be written as a product of an even number of transpositions. Thus, in particular, every product of k 3-cycles is a product of 2k transpositions and therefore in An. So whatever the 3-cycles generate must be a subgroup of An too.