Editing Symmetric group (section)

== Cyclic subgroups ==

[[Cyclic group]]s are those that are generated by a single permutation. When a permutation is represented in cycle notation, the order of the cyclic subgroup that it generates is the [[least common multiple]] of the lengths of its cycles.  For example, in S{{sub|5}}, one cyclic subgroup of order 5 is generated by (13254), whereas the largest cyclic subgroups of S{{sub|5}} are generated by elements like (123)(45) that have one cycle of length 3 and another cycle of length 2. This rules out many groups as possible subgroups of symmetric groups of a given size.{{cn|date=October 2022}} For example, S{{sub|5}} has no subgroup of order 15 (a divisor of the order of S{{sub|5}}), because the only group of order 15 is the cyclic group.  The largest possible order of a cyclic subgroup (equivalently, the largest possible order of an element in S{{sub|''n''}}) is given by [[Landau's function]].