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.
Link to originalThe 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