Editing P-group (section)

==Classification==
The groups of order ''p''<sup>''n''</sup> for 0 ≤ ''n'' ≤ 4 were classified early in the history of group theory,<ref>{{harv|Burnside|1897}}</ref> and modern work has extended these classifications to groups whose order divides ''p''<sup>7</sup>, though the sheer number of families of such groups grows so quickly that further classifications along these lines are judged difficult for the human mind to comprehend.<ref>{{harv|Leedham-Green|McKay|2002|p=214}}</ref> For example, [[Marshall Hall (mathematician)|Marshall Hall Jr.]] and James K. Senior classified groups of order 2<sup>''n''</sup> for ''n'' ≤ 6 in 1964.<ref>{{Harv|Hall Jr.|Senior|1964}}</ref>

Rather than classify the groups by order, [[Philip Hall]] proposed using a notion of [[isoclinism of groups]] which gathered finite ''p''-groups into families based on large quotient and subgroups.<ref>{{harv|Hall|1940}}</ref>

An entirely different method classifies finite ''p''-groups by their '''[[coclass]]''', that is, the difference between their [[composition series|composition length]] and their [[nilpotent group|nilpotency class]]. The so-called '''[[coclass conjectures]]''' described the set of all finite ''p''-groups of fixed coclass as perturbations of finitely many [[pro-p group]]s. The coclass conjectures were proven in the 1980s using techniques related to [[Lie algebra]]s and [[powerful p-group]]s.<ref>{{harv|Leedham-Green|McKay|2002}}</ref> The final proofs of the '''coclass theorems''' are due to A. Shalev and independently to C. R. Leedham-Green, both in 1994. They admit a classification of finite ''p''-groups in [[Descendant tree (group theory)#Multifurcation and coclass graphs|directed coclass graphs]] consisting of only finitely many coclass trees whose (infinitely many) members are characterized by finitely many parametrized presentations.

Every group of order ''p''<sup>5</sup> is [[Metabelian group|metabelian]].<ref name="metabelian">{{cite web | url=https://math.stackexchange.com/q/124010/178864 | title=Every group of order ''p''<sup>5</sup> is metabelian | date=24 March 2012 | publisher=Stack Exchange |access-date=7 January 2016}}</ref>

===Up to ''p''<sup>3</sup>===
The trivial group is the only group of order one, and the cyclic group C<sub>''p''</sub> is the only group of order ''p''. There are exactly two groups of order ''p''<sup>2</sup>, both abelian, namely C<sub>''p''<sup>2</sup></sub> and C<sub>''p''</sub>&nbsp;×&nbsp;C<sub>''p''</sub>. For example, the cyclic group C<sub>4</sub> and the [[Klein four-group]] ''V''<sub>4</sub> which is C<sub>2</sub>&nbsp;×&nbsp;C<sub>2</sub> are both 2-groups of order 4.

There are three abelian groups of order ''p''<sup>3</sup>, namely C<sub>''p''<sup>3</sup></sub>, C<sub>''p''<sup>2</sup></sub>&nbsp;×&nbsp;C<sub>''p''</sub>, and C<sub>''p''</sub>&nbsp;×&nbsp;C<sub>''p''</sub>&nbsp;×&nbsp;C<sub>''p''</sub>. There are also two non-abelian groups.

For ''p''&nbsp;≠&nbsp;2, one is a semi-direct product of C<sub>''p''</sub>&nbsp;×&nbsp;C<sub>''p''</sub> with C<sub>''p''</sub>, and the other is a semi-direct product of C<sub>''p''<sup>2</sup></sub> with C<sub>''p''</sub>. The first one can be described in other terms as group UT(3,''p'') of unitriangular matrices over finite field with ''p'' elements, also called the [[Heisenberg group#Heisenberg group modulo an odd prime p|Heisenberg group mod ''p'']].

For ''p''&nbsp;=&nbsp;2, both the semi-direct products mentioned above are isomorphic to the [[dihedral group]] Dih<sub>4</sub> of order 8. The other non-abelian group of order 8 is the [[quaternion group]] Q<sub>8</sub>.