Editing P-group (section)

==Examples==
''p''-groups of the same order are not necessarily [[isomorphism|isomorphic]]; for example, the [[cyclic group]] ''C''<sub>4</sub> and the [[Klein four-group]] ''V''<sub>4</sub> are both 2-groups of order 4, but they are not isomorphic.

Nor need a ''p''-group be [[abelian group|abelian]]; the [[dihedral group]] Dih<sub>4</sub> of order 8 is a non-abelian 2-group. However, every group of order ''p''<sup>2</sup> is abelian.<ref group="note">To prove that a group of order ''p''<sup>2</sup> is abelian, note that it is a ''p''-group so has non-trivial center, so given a non-trivial element of the center ''g,'' this either generates the group (so ''G'' is cyclic, hence abelian: <math>G=C_{p^2}</math>), or it generates a subgroup of order ''p,'' so ''g'' and some element ''h'' not in its orbit generate ''G,'' (since the subgroup they generate must have order <math>p^2</math>) but they commute since ''g'' is central, so the group is abelian, and in fact <math>G=C_p \times C_p.</math></ref>

The dihedral groups are both very similar to and very dissimilar from the [[quaternion group]]s and the [[semidihedral group]]s. Together the dihedral, semidihedral, and quaternion groups form the 2-groups of [[maximal class]], that is those groups of order 2<sup>''n''+1</sup> and nilpotency class ''n''.

===Iterated wreath products===
The iterated [[wreath product]]s of cyclic groups of order ''p'' are very important examples of ''p''-groups. Denote the cyclic group of order ''p'' as ''W''(1), and the wreath product of ''W''(''n'') with ''W''(1) as ''W''(''n''&nbsp;+&nbsp;1). Then ''W''(''n'') is the Sylow ''p''-subgroup of the [[symmetric group]] Sym(''p''<sup>''n''</sup>). Maximal ''p''-subgroups of the general linear group GL(''n'','''Q''') are direct products of various ''W''(''n''). It has order ''p''<sup>''k''</sup> where ''k''&nbsp;=&nbsp;(''p''<sup>''n''</sup>&nbsp;−&nbsp;1)/(''p''&nbsp;−&nbsp;1). It has nilpotency class ''p''<sup>''n''−1</sup>, and its lower central series, upper central series, lower exponent-''p'' central series, and upper exponent-''p'' central series are equal. It is generated by its elements of order ''p'', but its exponent is ''p''<sup>''n''</sup>. The second such group, ''W''(2), is also a ''p''-group of maximal class, since it has order ''p''<sup>''p''+1</sup> and nilpotency class ''p'', but is not a [[regular p-group|regular ''p''-group]]. Since groups of order ''p''<sup>''p''</sup> are always regular groups, it is also a minimal such example.

===Generalized dihedral groups===
When ''p''&nbsp;=&nbsp;2 and ''n''&nbsp;=&nbsp;2, ''W''(''n'') is the dihedral group of order 8, so in some sense ''W''(''n'') provides an analogue for the dihedral group for all primes ''p'' when ''n''&nbsp;=&nbsp;2. However, for higher ''n'' the analogy becomes strained. There is a different family of examples that more closely mimics the dihedral groups of order 2<sup>''n''</sup>, but that requires a bit more setup. Let ζ denote a primitive ''p''th root of unity in the complex numbers, let '''Z'''[ζ] be the ring of [[ring of integers|cyclotomic integers]] generated by it, and let ''P'' be the [[prime ideal]] generated by 1−ζ. Let ''G'' be a cyclic group of order ''p'' generated by an element ''z''. Form the [[semidirect product]] ''E''(''p'') of '''Z'''[ζ] and ''G'' where ''z'' acts as multiplication by ζ. The powers ''P''<sup>''n''</sup> are normal subgroups of ''E''(''p''), and the example groups are ''E''(''p'',''n'')&nbsp;=&nbsp;''E''(''p'')/''P''<sup>''n''</sup>. ''E''(''p'',''n'') has order ''p''<sup>''n''+1</sup> and nilpotency class ''n'', so is a ''p''-group of maximal class. When ''p''&nbsp;=&nbsp;2, ''E''(2,''n'') is the dihedral group of order 2<sup>''n''</sup>. When ''p'' is odd, both ''W''(2) and ''E''(''p'',''p'') are irregular groups of maximal class and order ''p''<sup>''p''+1</sup>, but are not isomorphic.

===Unitriangular matrix groups===

The Sylow subgroups of [[general linear group]]s are another fundamental family of examples. Let ''V'' be a vector space of dimension ''n'' with basis { ''e''<sub>1</sub>, ''e''<sub>2</sub>, ..., ''e''<sub>''n''</sub> } and define ''V''<sub>''i''</sub> to be the vector space generated by { ''e''<sub>''i''</sub>, ''e''<sub>''i''+1</sub>, ..., ''e''<sub>''n''</sub> } for 1 ≤ ''i'' ≤ ''n'', and define ''V''<sub>''i''</sub> = 0 when ''i'' &gt; ''n''. For each 1 ≤ ''m'' ≤ ''n'', the set of invertible linear transformations of ''V'' which take each ''V''<sub>''i''</sub> to ''V''<sub>''i''+''m''</sub> form a subgroup of Aut(''V'') denoted ''U''<sub>''m''</sub>. If ''V'' is a vector space over '''Z'''/''p'''''Z''', then ''U''<sub>1</sub> is a Sylow ''p''-subgroup of Aut(''V'') = GL(''n'', ''p''), and the terms of its [[lower central series]] are just the ''U''<sub>''m''</sub>. In terms of matrices, ''U''<sub>''m''</sub> are those upper triangular matrices with 1s one the diagonal and 0s on the first ''m''−1 superdiagonals. The group ''U''<sub>1</sub> has order ''p''<sup>''n''·(''n''−1)/2</sup>, nilpotency class ''n'', and exponent ''p''<sup>''k''</sup> where ''k'' is the least integer at least as large as the base ''p'' [[logarithm]] of ''n''.