Editing List of small groups (section)

== Glossary ==
Each group is named by [[#Small Groups Library|Small Groups library]] as G<sub>''o''</sub><sup>''i''</sup>, where ''o'' is the order of the group, and ''i'' is the  index used to label the group within that order.

Common group names:
* Z<sub>''n''</sub>: the [[cyclic group]] of order ''n'' (the notation C<sub>''n''</sub> is also used; it is isomorphic to the [[additive group]] of '''Z'''/''n'''''Z''')
* Dih<sub>''n''</sub>: the [[dihedral group]] of order 2''n'' (often the notation D<sub>''n''</sub> or D<sub>2''n''</sub> is used)
** K<sub>4</sub>: the [[Klein four-group]] of order 4, same as {{nowrap|Z<sub>2</sub> × Z<sub>2</sub>}} and Dih<sub>2</sub>
* D<sub>2''n''</sub>: the dihedral group of order 2''n'', the same as Dih<sub>''n''</sub> (notation used in section [[#List of small non-abelian groups|List of small non-abelian groups]])
* S<sub>''n''</sub>: the [[symmetric group]] of degree ''n'', containing the [[factorial|''n''!]] [[permutation]]s of ''n'' elements
* A<sub>''n''</sub>: the [[alternating group]] of degree ''n'', containing the [[even permutation]]s of ''n'' elements, of order 1 for {{nowrap|1=''n'' = 0,&thinsp;1}}, and order ''n''!/2 otherwise
* Dic<sub>''n''</sub> or Q<sub>4''n''</sub>: the [[dicyclic group]] of order 4''n''
** Q<sub>8</sub>: the [[quaternion group]] of order 8, also Dic<sub>2</sub>

The notations Z<sub>''n''</sub> and Dih<sub>''n''</sub> have the advantage that [[point groups in three dimensions]] C<sub>''n''</sub> and D<sub>''n''</sub> do not have the same notation. There are more [[isometry group]]s than these two, of the same abstract group type.

The notation {{nowrap|''G'' × ''H''}} denotes the [[direct product of groups|direct product]] of the two groups; ''G''<sup>''n''</sup> denotes the direct product of a group with itself ''n'' times. ''G'' ⋊ ''H'' denotes a [[semidirect product]] where ''H'' [[group action|acts]] on ''G''; this may also depend on the choice of action of ''H'' on ''G''.

[[Abelian group|Abelian]] and [[simple group]]s are noted. (For groups of order {{nowrap|''n'' < 60}}, the simple groups are precisely the cyclic groups Z<sub>''n''</sub>, for [[prime number|prime]] ''n''.) The equality sign ("=") denotes isomorphism.

The [[identity element]] in the [[cycle graph (algebra)|cycle graphs]] is represented by the black circle.  The lowest order for which the cycle graph does not uniquely represent a group is order 16.

In the lists of [[subgroup]]s, the trivial group and the group itself are not listed.  Where there are several isomorphic subgroups, the number of such subgroups is indicated in parentheses.

[[Angle brackets]] <relations> show the [[presentation of a group]].