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Answer: | Answer: | ||
| − | Suppose H contains at least one odd permutation, say <math>\sigma</math>. For each odd permutation <math>\beta</math>, the permutation <math>\sigma \beta</math> | + | Suppose H contains at least one odd permutation, say <math>\sigma</math>. For each odd permutation <math>\beta</math>, the permutation <math>\sigma \beta</math> is even. |
| + | |||
| + | Note: | ||
| + | <math>\sigma</math> = odd | ||
| + | <math>\beta</math> = odd | ||
| + | <math>\sigma \beta </math> = even | ||
Revision as of 15:24, 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
