Lemma·L36
Raising to the group order gives the identity
In a finite group, every element raised to the number of group elements is the identity.
In words
In a finite group, for every g, if g is an element of the group then applying g to itself as many times as there are elements in the group brings you back to the identity.
Never needed: F10 · A03 · A04 · A05 · A07 (computed from the citation graph, not asserted).
Proof
- 1
- 2
∎
Remarks
Specialised to the multiplicative group of nonzero residues modulo a prime, this is Fermat's little theorem
; modulo a general
it is Euler's theorem. Both statements await modular arithmetic, which needs the integers and quotient rings, not yet constructed in this library. The proof uses only that the order of
divides
(T15).