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Math and science::Algebra::Aluffi

Groups. All orders divides order of maximal element

Order: divides order of maximal element

Let \( G \) be a [something] group, and let \( g \in G \) be an element with maximal finite order. Any other element with finite order has an order that [has what relationship?].

The same statement cannot be made for non-comutative groups.

Can you remember the proof?

Maximal finite order

Let \( g \) be an element of group \( G \). \( g \) has maximal finite order iff all other elements with finite order have an order less than that of \( g \).