Editing Parity (mathematics) (section)

===Group theory===
[[File:Rubiks revenge solved.jpg|thumb|left|[[Rubik's Revenge]] in solved state]]
The [[parity of a permutation]] (as defined in [[abstract algebra]]) is the parity of the number of [[Transposition (mathematics)|transposition]]s into which the permutation can be decomposed.<ref>{{citation|title=Permutation Groups|volume=45|series=London Mathematical Society Student Texts|first=Peter J.|last=Cameron|author-link=Peter Cameron (mathematician)|publisher=Cambridge University Press|year=1999|isbn=9780521653787|pages=26–27|url=https://books.google.com/books?id=4bNj8K1omGAC&pg=PA26}}.</ref> For example (ABC) to (BCA) is even because it can be done by swapping A and B then C and A (two transpositions). It can be shown that no permutation can be decomposed both in an even and in an odd number of transpositions. Hence the above is a suitable definition. In [[Rubik's Cube]], [[Megaminx]], and other twisting puzzles, the moves of the puzzle allow only even permutations of the puzzle pieces, so parity is important in understanding the [[Configuration space (mathematics)|configuration space]] of these puzzles.<ref>{{citation|title=Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys|first=David|last=Joyner|publisher=JHU Press|year=2008|isbn=9780801897269|contribution=13.1.2 Parity conditions|pages=252–253|url=https://books.google.com/books?id=iM0fco-_Ri8C&pg=PA252}}.</ref>

The [[Feit&ndash;Thompson theorem]] states that a [[finite group]] is always solvable if its order is an odd number.  This is an example of odd numbers playing a role in an advanced mathematical theorem where the method of application of the simple hypothesis of "odd order" is far from obvious.<ref>{{citation
 | last1 = Bender | first1 = Helmut
 | last2 = Glauberman | first2 = George
 | isbn = 978-0-521-45716-3
 | location = Cambridge
 | mr = 1311244
 | publisher = Cambridge University Press
 | series = London Mathematical Society Lecture Note Series
 | title = Local analysis for the odd order theorem
 | volume = 188
 | year = 1994}}; {{citation
 | last = Peterfalvi | first = Thomas
 | isbn = 978-0-521-64660-4
 | location = Cambridge
 | mr = 1747393
 | publisher = Cambridge University Press
 | series = London Mathematical Society Lecture Note Series
 | title = Character theory for the odd order theorem
 | volume = 272
 | year = 2000}}.</ref>