/sci/ - Science & Math

A good morning /mg/!

>>11627031
(10) Like I mentioned before, 6=3! makes every composite the identity. In this case it is also the smallest option.
(13) It's worth checking that it really does the trick. Just remember that all permutations are just products/composites of cycles of different length and then you want to kill all the cycles.
These things you are verifying now for these symmetry groups will be discussed more generally later. The cardinality of the group as a set is called its order, and the smallest positive integer (or infinity) turning an element to the identity element is called the order of the element. Later in the book, Herstein will show that the order of an element divides the order of the group in the finite case, but this does not work the other way around though. There are, like you wrote them down, only those 6 possible permutations of 3 elements, but none of them has order 6. Just something to keep in mind when you get there!