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Calculus and Beyond Homework Help
Showing that an element has order 2 if product of 2-cycles
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[QUOTE="fresh_42, post: 6019196, member: 572553"] Yes, but only implicit in the word "cycle decomposition". Since you haven't filled out part two of the template which I personally consider even more important than part three, although our rules lay emphasis on own effort, you have to rely on the fact that everybody knows, that ##(12)(23)## isn't a decomposition whereas ##(123)## is. I think you should have mentioned this somewhere because it is crucial for the proof. Both directions are correct, since you already have proven ##|\sigma \tau|=\operatorname{lcm}\{\,|\sigma | , |\tau | \,\}## in case ##[\sigma,\tau]=1## and disjoint permutations commute. However, both facts are necessary for your way to prove it - plus a proper definition of a cycle decomposition. You see, there are many hidden statements, which you all assumed to be known. [/QUOTE]
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Showing that an element has order 2 if product of 2-cycles
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