Answer to Question #156046 in Abstract Algebra for arci azarcon

Question #156046

1.   Prove that a cycle of length l is odd if l is even.


1
Expert's answer
2021-01-19T01:56:36-0500

Let "(a_1a_2...a_l)" be a cycle of even length "l". Taking into account that "(a_1a_2...a_l)=(a_1a_l)(a_1a_{l-1})...(a_1a_3)(a_1a_2)", we conclude that this cycle is represented as a product of "l-1" transpositions. Since "l" is even, "l-1" is odd, and thus "(a_1a_2...a_l)" is odd.


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