Definition·D044
Order of a group element
The least positive number of times an element must be applied to itself to return to the identity.
Let
be a group with
finite, and
. The order of
is the least positive natural at which its power is the identity:
with powers from D043,
from D026, and
from D029.
In words
The order of g is the least positive natural n with g to the n equal to the identity.
Rests onA02
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
Such an
exists by L33, and 'the least' is well defined by the well-ordering principle applied to the nonempty set
(Separation). The identity is the unique element of order
. The order equals the size of the cyclic subgroup
, and by T15 it divides
. We define the order only for finite groups, where L33 guarantees existence; in an infinite group an element may have no finite order.