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| + | The problem asks us to show that <math>A_8</math> contains an element of order 15. | ||
| + | Now bear with me, because I feel like I'm only on the cusp of understanding this stuff, but I think we can show it this way: | ||
| + | |||
| + | We have 8 elements: | ||
| + | |||
| + | '''1 2 3 4 5 6 7 8''' | ||
| + | |||
| + | that we can arrange into cycles 3 and 5: | ||
| + | |||
| + | '''(123)(45678)'''. | ||
| + | |||
| + | Now since the lcm of 3 and 5 is 15, the order of this element is 15, from pg 101 in the book. | ||
| + | We can verify that this element is, in fact, in <math>A_8</math> by examining the transpositions: | ||
| + | |||
| + | '''(123)''' --> '''(12)(13)''' = 2 | ||
| + | |||
| + | '''(45678)''' --> '''(45)(46)(47)(48)''' = 4 | ||
| + | |||
| + | Since 4 + 2 = 6 is even, this element belongs to <math>A_8</math>. | ||
| + | |||
| + | QED? Please correct me if I'm doing this wrong. | ||
| + | |||
| + | -Tim | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | This also works if you follow Anna's suggestion for problem #8: 5 and 3 are the logical primes which add to 8 and whose lcm is 15. | ||
Revision as of 17:26, 10 September 2008
The problem asks us to show that $ A_8 $ contains an element of order 15.
Now bear with me, because I feel like I'm only on the cusp of understanding this stuff, but I think we can show it this way:
We have 8 elements:
1 2 3 4 5 6 7 8
that we can arrange into cycles 3 and 5:
(123)(45678).
Now since the lcm of 3 and 5 is 15, the order of this element is 15, from pg 101 in the book. We can verify that this element is, in fact, in $ A_8 $ by examining the transpositions:
(123) --> (12)(13) = 2
(45678) --> (45)(46)(47)(48) = 4
Since 4 + 2 = 6 is even, this element belongs to $ A_8 $.
QED? Please correct me if I'm doing this wrong.
-Tim
This also works if you follow Anna's suggestion for problem #8: 5 and 3 are the logical primes which add to 8 and whose lcm is 15.
