Group

Criterion for in a group

For a group element , iff either: group

• and
• and

Proof

If has infinite order there exists no nonzero such that , and since implies , it follows .

If then we again have the implication . By the division algorithm , with . Then , and since is the lowest positive integer such that , it follows that . Hence .

Corollary

It immediately follows that implies divides .

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