Theorem·T15
The order of an element divides the order of the group
In a finite group, the order of every element divides the number of elements.
Let
be a group with
finite. The order of every element divides the order of the group:
with
from D030 and
from D035.
In words
Let G with operation ⋆ be a finite group. For every g, if g is an element of G then the order of g divides the number of elements of G.
Never needed: F10 · A03 · A04 · A05 · A07 (computed from the citation graph, not asserted).
Proof
- 1Fix . By L35 the cyclic subgroup satisfies .
- 2By Lagrange's theorem applied to the subgroup , . Substituting (substitutivity) gives .
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Remarks
This is the most-used corollary of Lagrange's theorem. It immediately bounds the possible orders of elements and, via L36, gives
for every
, the group form of Fermat's and Euler's theorems. It also drives the classification of prime-order groups.