Editing P-group (section)

===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.