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by Ciro Santilli (@cirosantilli, 37)

Cycle of an element of a group

 Home Mathematics Area of mathematics Algebra Group Generating set of a group
 0 By others on same topic  0 Discussions  Updated 2025-05-29  +Created 1970-01-01  See my version
Take the element and apply it to itself. Then again. And so on.
In the case of a finite group, you have to eventually reach the identity element again sooner or later, giving you the order of an element of a group.
The continuous analogue for the cycle of a group are the one parameter subgroups. In the continuous case, you sometimes reach identity again and to around infinitely many times (which always happens in the finite case), but sometimes you don't.
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    • Order of an element of a group Cycle of an element of a group

Order of an element of a group

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Cycle of an element of a group
The length of its cycle.
Bibliography:
  • math.stackexchange.com/questions/972057/calculating-the-order-of-an-element-in-a-group

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