2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/112952In this article, we introduce the notion of cycling operations of arbitrary order in Garside groups, which is a full generalization of the cycling and decycling operations. Theoretically, this notion together with other related concepts provides a context in which various definitions and arguments concerning Garside groups are unified and simplified as well as improved. Practically, it yields a new algorithm which has a considerably improved performance on solving the conjugacy problem of reducible braids.22 pages. A new section (Section 7) was appended in the second version, in which we present several applications on the summit infimum/supremum, the stable summit set and the rigid elementsGeometric TopologyGroup Theory20F36; 20F10text