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Note: | Note: | ||
<math>\sigma</math> = odd | <math>\sigma</math> = odd | ||
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<math>\beta</math> = odd | <math>\beta</math> = odd | ||
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<math>\sigma \beta </math> = even | <math>\sigma \beta </math> = even | ||
Revision as of 15:25, 9 September 2008
Question: Show that if H is a subgroup of $ S_n $, then either every member of H is an even permutation or exactly half of the members are even.
Answer: Suppose H contains at least one odd permutation, say $ \sigma $. For each odd permutation $ \beta $, the permutation $ \sigma \beta $ is even.
Note: $ \sigma $ = odd
$ \beta $ = odd
$ \sigma \beta $ = even
