Lemma·L33
Elements of a finite group have a finite order
In a finite group some positive power of every element is the identity.
Let
be a group with
finite, and
. Then some positive power of
is the identity:
with powers from D043 and
from D029.
In words
If the group has only finitely many elements and g is any of its elements, then there is an exponent n: n is a natural number, and n is positive, and multiplying g by itself n times returns to the identity.
Never needed: F10 · A03 · A04 · A05 · A07 (computed from the citation graph, not asserted).
Proof
- 1Let (D035) and fix a bijection (D034). Let be the restriction of (D043) to , so for each ; these are the powers .
- 2
- 3
- 4
∎
Remarks
The heart is pigeonhole: among the
powers
two must coincide in a group of
elements, and dividing them out yields a power equal to
. This is exactly what makes the order of an element well defined. In an infinite group the conclusion can fail: the integers under addition, once constructed, have elements of no finite order.