Group order

Order of powers of a group element

Given a group element with , then and . group

Proof

Let and . Since by closure . By Bézout’s lemma, there exist such that , so that . Hence by closure and therefore .

Keeping in mind is a divisor of , it is clear that , implying . But if then and therefore by the definition of order. Hence .

Using this technique, computing the cyclic group generated by some power of a basic element becomes simple.¹

Corollaries

Order of elements in finite cyclic groups

It immediately follows that the order of an element in a finite cyclic group divides the order of the group. group

Criterion for ‹𝑎ⁱ› = ‹𝑎ʲ› and |𝑎ⁱ| = |𝑎ʲ| in a group

Given a group element with , then iff . Likewise iff . group

Proof

From the above theorem, iff . Clearly implies . On the other hand, implies and thence .

It follows immediately that implies . From the above theorem, .

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Footnotes

1. 2017, Contemporary Abstract Algebra, p. 79 ↩