Editing P-group (section)

==Prevalence==

=== Among groups ===
{{main article|Higman–Sims asymptotic formula}}

The [[Higman–Sims asymptotic formula]] states that the number of isomorphism classes of groups of order ''p<sup>n</sup>'' grows as <math>p^{\frac{2}{27}n^3+O(n^{8/3})}</math>, and these are dominated by the classes that are two-step nilpotent.<ref>{{harv|Sims|1965}}</ref> Because of this rapid growth, there is a [[Mathematical folklore|folklore]] conjecture asserting that almost all [[finite group]]s are 2-groups: the fraction of [[isomorphism class]]es of 2-groups among isomorphism classes of groups of order at most ''n'' is thought to tend to 1 as ''n'' tends to infinity. For instance, of the 49 910 529 484 different groups of order at most 2000, {{gaps|49|487|367|289}}, or just over 99%, are 2-groups of order 1024.<ref name="Burrell">{{cite journal |last1=Burrell |first1=David |title=On the number of groups of order 1024 |journal=Communications in Algebra |date=2021-12-08 |volume=50 |issue=6 |pages=2408–2410 |doi=10.1080/00927872.2021.2006680 |url=https://www.tandfonline.com/doi/full/10.1080/00927872.2021.2006680}}</ref>

=== Within a group ===
Every finite group whose order is divisible by ''p'' contains a subgroup which is a non-trivial ''p''-group, namely a cyclic group of order ''p'' generated by an element of order ''p'' obtained from [[Cauchy's theorem (group theory)|Cauchy's theorem]]. In fact, it contains a ''p''-group of maximal possible order: if <math>|G|=n=p^km</math> where ''p'' does not divide ''m,'' then ''G'' has a subgroup ''P'' of order <math>p^k,</math> called a Sylow ''p''-subgroup. This subgroup need not be unique, but any subgroups of this order are conjugate, and any ''p''-subgroup of ''G'' is contained in a Sylow ''p''-subgroup. This and other properties are proved in the [[Sylow theorems]].