Cyclic permutation of factors preserves order of elements

abstract-algebra cyclic-groups permutations order-theory

Cyclic permutations arise by conjugation, e.g. $\,a_3 a_4\cdots a_n\,a_1a_2 = (a_1 a_2)^{-1}(a_1 a_2 \cdots a_n) (a_1 a_2)$

But generally conjugation $\ g\mapsto a^{-1}ga,\, $ is a group isomorphism, with inverse $\ g\mapsto aga^{-1}.$ Isomorphisms preserve all "group-theoretic" properties, which includes the order of an element $\,g,\,$ since this equals the order (cardinality) of the cyclic group generated by $\,g.\,$ But an isomorphic image of a group has the same order (cardinality).

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