See Modulo Multiplication Group: https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n http://mathworld.wolfram.com/ModuloMultiplicationGroup.html
Since 11 is prime, its modulo multiplication group has 10 elements (all the numbers from 1 to 10). i.e. multiplying the elements by any number in 1 to 10 will permute the elements. If you used 8, say, instead. 8 is only coprime to 1,3,5,7. So its Modulo multiplication group has only 4 elements. If you do 8 | 2 * i.8 you do not get a permutation of the numbers of i.8, because 2 and 8 are not coprime. A long time ago I wrote a script to calculate the modulo multiplication group for arbitrary integers. https://github.com/jonghough/PermuJ/blob/master/modulomultiplication.ijs On Wednesday, January 29, 2020, 10:20:12 AM GMT+9, Jimmy Gauvin <jimmy.gau...@gmail.com> wrote: Hi, I am looking for some reference texts on permutations and modular arithmetic. I recently stumbled on some interesting properties of card shuffles. For example, using a deck of 11 cards labeled 0 through 10 and shuffling them to obtain this layout : 0 6 1 7 2 8 3 9 4 10 There are several ways to find out which position each card occupies . 1) index of 0 6 1 7 2 8 3 9 4 10 i. i.11 0 2 4 6 8 10 1 3 5 7 9 2) grading /: 0 6 1 7 2 8 3 9 4 10 0 2 4 6 8 10 1 3 5 7 9 3) and computing the positions with modulo 11 | 2*i.11 0 2 4 6 8 10 1 3 5 7 9 Going from the positions to the card layout can also be done several ways : 4) assignment (i.11) ( 0 2 4 6 8 10 1 3 5 7 9 ) } 11$0 0 6 1 7 2 8 3 9 4 10 5) grading /: 0 2 4 6 8 10 1 3 5 7 9 0 6 1 7 2 8 3 9 4 10 6) and, this is the kicker for me, modulo with the right multiplier 11 | 6*i.11 0 6 1 7 2 8 3 9 4 10 While 3) is obvious, I find 6) disconcerting. And it seems to work for all cases where the number of cards and the interval between cards are coprime. I know this must be explained somewhere but I can't find the relevant material. Thanks for your assistance, Jimmy ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
