Potentiation in a group:

Potentiation for a group works just as we are used to from highschool:

Note: It’s to turn the negative into a positive number.

Theorem: ,

In the additive group , potentiation becomes repeated addition (= multiplication) rather than repeated multiplication:

In an abelian group (additive context), we write instead of and instead of .

Why is ?


And ? If you are working in areas like combinatorics, set theory, abstract algebra, or computer science, is generally taken to be . This definition simplifies many formulas and theorems.

The set of all powers of :

Let be a group and , then , is a cyclic subgroup of , generated by (“von erzeugte Untergruppe”).

, … is it actually a subgroup?
Note empty:
Closed: abelian group
Invertible: (because

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