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.
