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.

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