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| + | Problem Statement: | ||
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| + | Show that <math>A_8</math> contains an element of order 15 | ||
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The problem asks us to show that <math>A_8</math> contains an element of order 15. | The problem asks us to show that <math>A_8</math> contains an element of order 15. | ||
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-Tim | -Tim | ||
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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. | 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. | ||
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| + | [[HW2_MA453Fall2008walther|Back to HW2]] | ||
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| + | [[Main_Page_MA453Fall2008walther|Back to MA453 Fall 2008 Prof. Walther]] | ||
Latest revision as of 16:33, 22 October 2010
HW2, Chapter 5 problem 6, Discussion, MA453, Prof. Walther
Problem Statement:
Show that $ A_8 $ contains an element of order 15
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.
