(New page: Here's a hint (that I found helpful): Consider an element of A_{10} as a permutation written in disjoint cycle notation. The lengths of the cycles must add up to no more than 10, since t...) |
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| − | + | =[[HW2_MA453Fall2008walther|HW2]], Chapter 5 problem 6, Discussion, [[MA453]], [[user:walther|Prof. Walther]]= | |
| − | + | 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. | ||
| + | |||
| + | 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: | ||
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| + | We have 8 elements: | ||
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| + | '''1 2 3 4 5 6 7 8''' | ||
| + | |||
| + | that we can arrange into cycles 3 and 5: | ||
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| + | '''(123)(45678)'''. | ||
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| + | 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: | ||
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| + | '''(123)''' --> '''(12)(13)''' = 2 | ||
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| + | '''(45678)''' --> '''(45)(46)(47)(48)''' = 4 | ||
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| + | Since 4 + 2 = 6 is even, this element belongs to <math>A_8</math>. | ||
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| + | QED? Please correct me if I'm doing this wrong. | ||
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| + | -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. | ||
| + | ---- | ||
| + | [[HW2_MA453Fall2008walther|Back to HW2]] | ||
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| + | [[Main_Page_MA453Fall2008walther|Back to MA453 Fall 2008 Prof. Walther]] | ||
Latest revision as of 15:33, 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.
