Editing Order (group theory) (section)

==Order and structure==
The order of a group ''G'' and the orders of its elements give much information about the structure of the group. Roughly speaking, the more complicated the [[factorization]] of |''G''|, the more complicated the structure of ''G''.

For  |''G''| = 1, the group is [[trivial group|trivial]]. In any group, only the identity element ''a = e'' has ord(''a)'' = 1. If every non-identity element in ''G'' is equal to its inverse (so that ''a''<sup>2</sup> = ''e''), then ord(''a'') = 2; this implies ''G'' is [[abelian group|abelian]] [[group theory#Inverse of ab|since]] <math>ab=(ab)^{-1}=b^{-1}a^{-1}=ba</math>. The converse is not true; for example, the (additive) [[cyclic group]] '''Z'''<sub>6</sub> of integers [[Modular arithmetic|modulo]] 6 is abelian, but the number 2 has order 3:

:<math>2+2+2=6 \equiv 0 \pmod {6}</math>.

The relationship between the two concepts of order is the following: if we write
:<math>\langle a \rangle = \{ a^{k}\colon k \in \mathbb{Z} \} </math>
for the [[subgroup]] [[Generating set of a group|generated]] by ''a'', then
:<math>\operatorname{ord} (a) = \operatorname{ord}(\langle a \rangle).</math>

For any integer ''k'', we have
:''a<sup>k</sup>'' = ''e'' &nbsp; if and only if &nbsp; ord(''a'') [[divisor|divides]] ''k''.

In general, the order of any subgroup of ''G'' divides the order of ''G''. More precisely: if ''H'' is a subgroup of ''G'', then
:ord(''G'') / ord(''H'') = [''G'' : ''H''], where [''G'' : ''H''] is called the [[index of a subgroup|index]] of ''H'' in ''G'', an integer. This is [[Lagrange's theorem (group theory)|Lagrange's theorem]]. (This is, however, only true when G has finite order. If ord(''G'') = ∞, the quotient ord(''G'') / ord(''H'') does not make sense.)

As an immediate consequence of the above, we see that the order of every element of a group divides the order of the group. For example, in the symmetric group shown above, where ord(S<sub>3</sub>)&nbsp;= 6, the possible orders of the elements are 1, 2, 3 or 6.

The following partial converse is true for [[finite group]]s: if ''d'' divides the order of a group ''G'' and ''d'' is a [[prime number]], then there exists an element of order ''d'' in ''G'' (this is sometimes called [[Cauchy's theorem (group theory)|Cauchy's theorem]]). The statement does not hold for [[composite number|composite]] orders, e.g. the [[Klein four-group]] does not have an element of order four. This can be shown by [[inductive proof]].<ref>{{Cite web|title=Proof of Cauchy's Theorem|url=http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/cauchypf.pdf|first=Keith|last=Conrad|format=PDF|access-date=May 14, 2011|url-status=dead|archive-url=https://web.archive.org/web/20181123110229/http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/cauchypf.pdf|archive-date=2018-11-23}}</ref> The consequences of the theorem include: the order of a group ''G'' is a power of a prime ''p'' if and only if ord(''a'') is some power of ''p'' for every ''a'' in ''G''.<ref>{{Cite web|title=Consequences of Cauchy's Theorem|url=http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/cauchyapp.pdf|first=Keith|last=Conrad|format=PDF|access-date=May 14, 2011|url-status=dead|archive-url=https://web.archive.org/web/20180712201823/http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/cauchyapp.pdf|archive-date=2018-07-12}}</ref>

If ''a'' has infinite order, then all non-zero powers of ''a'' have infinite order as well. If ''a'' has finite order, we have the following formula for the order of the powers of ''a'':
:ord(''a<sup>k</sup>'') = ord(''a'') / [[greatest common divisor|gcd]](ord(''a''), ''k'')<ref>Dummit, David; Foote, Richard. ''Abstract Algebra'', {{isbn|978-0471433347}}, pp. 57</ref>
for every integer ''k''. In particular, ''a'' and its inverse ''a''<sup>−1</sup> have the same order.

In any group, 
:<math> \operatorname{ord}(ab) = \operatorname{ord}(ba)</math>

There is no general formula relating the order of a product ''ab'' to the orders of ''a'' and ''b''. In fact, it is possible that both ''a'' and ''b'' have finite order while ''ab'' has infinite order, or that both ''a'' and ''b'' have infinite order while ''ab'' has finite order. An example of the former is ''a''(''x'') = 2−''x'', ''b''(''x'') = 1−''x'' with ''ab''(''x'') = ''x''−1 in the group <math>Sym(\mathbb{Z})</math>. An example of the latter is ''a''(''x'') = ''x''+1, ''b''(''x'') = ''x''−1 with ''ab''(''x'') = ''x''. If ''ab'' = ''ba'', we can at least say that ord(''ab'') divides [[least common multiple|lcm]](ord(''a''), ord(''b'')). As a consequence, one can prove that in a finite abelian group, if ''m'' denotes the maximum of all the orders of the group's elements, then every element's order divides ''m''.