Editing Alternating group (section)

==Example: the 15 puzzle==
[[File:15-puzzle magical.svg|thumb|150px|A [[15 puzzle]].]]
It can be proved that the [[15 puzzle]], a famous example of the [[sliding puzzle]], can be represented by the alternating group A<sub>15</sub>,<ref>{{cite web |last1=Beeler |first1=Robert |title=The Fifteen Puzzle: A Motivating Example for the Alternating Group |url=https://faculty.etsu.edu/beelerr/fifteen-supp.pdf |website=faculty.etsu.edu/ |publisher=East Tennessee State University |access-date=2020-12-26 |archive-date=2021-01-07 |archive-url=https://web.archive.org/web/20210107214840/https://faculty.etsu.edu/beelerr/fifteen-supp.pdf |url-status=dead }}</ref> because the combinations of the 15 puzzle can be generated by [[Permutation#Definition|3-cycles]]. In fact, any {{nowrap|2''k'' − 1}} sliding puzzle with square tiles of equal size can be represented by A<sub>2''k''−1</sub>.