A cycle sends to , continuing cyclically back from to , and fixes elements outside its support. Products conventionally act from right to left. Every permutation is a product of disjoint cycles, unique up to their ordering and cyclic rotation of each displayed list.
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A permutation group can then be described in terms of the generating set of a group of specific elements given in cycle notation.
E.g. en.wikipedia.org/w/index.php?title=Mathieu_group&oldid=1034060469#Permutation_groups mentions that the Mathieu group is generated by three elements:which feels quite compact for a simple group with 95040 elements, doesn't it!
- (0123456789a)
- (0b)(1a)(25)(37)(48)(69)
- (26a7)(3945)