Editing Sylow theorems (section)

==Proof of the Sylow theorems==
The Sylow theorems have been proved in a number of ways, and the history of the proofs themselves is the subject of many papers, including Waterhouse,{{sfn|Waterhouse|1980}} Scharlau,{{sfn|Scharlau|1988}} Casadio and Zappa,{{sfn|Casadio|Zappa|1990}} Gow,{{sfn|Gow|1994}} and to some extent Meo.{{sfn|Meo|2004}}

One proof of the Sylow theorems exploits the notion of [[Group action (mathematics)|group action]] in various creative ways. The group {{mvar|G}} acts on itself or on the set of its ''p''-subgroups in various ways, and each such action can be exploited to prove one of the Sylow theorems.  The following proofs are based on combinatorial arguments of Wielandt.{{sfn|Wielandt|1959}} In the following, we use <math>a \mid b</math> as notation for "a divides b" and <math>a \nmid b</math> for the negation of this statement.

{{math theorem|note=1|1=A finite group {{var|G}} whose order <math>|G|</math> is divisible by a prime power ''p<sup>k</sup>'' has a subgroup of order ''p<sup>k</sup>''.}}

{{math proof|1=Let {{math|1={{abs|{{var|G}}}} = ''p<sup>k</sup>m'' = ''p''<sup>''k''+''r''</sup>''u''}} such that <math>p \nmid u</math>, and let Ω denote the set of subsets of {{mvar|G}} of size ''p<sup>k</sup>''. {{mvar|G}} [[Group action (mathematics)|acts]] on Ω by left multiplication: for {{math|1={{var|g}} ∈ {{var|G}}}} and {{math|1={{var|ω}} ∈ Ω}}, {{math|1={{var|g}}⋅{{var|ω}} = {{mset| {{var|g}}{{var|x}} {{pipe}} {{var|x}} ∈ {{var|ω}} }}}}. For a given set {{math|1={{var|ω}} ∈ Ω}}, write {{var|G}}<sub>{{var|ω}}</sub> for its [[Group action (mathematics)#Orbits and stabilizers|stabilizer subgroup]] {{math|1={{mset| {{var|g}} ∈ {{var|G}} {{pipe}} {{var|g}}⋅{{var|ω}} {{=}} {{var|ω}} }}}} and {{var|G}}{{var|ω}} for its [[Group action (mathematics)#Orbits and stabilizers|orbit]] {{math|1={{mset| {{var|g}}⋅{{var|ω}} {{pipe}} {{var|g}} ∈ {{var|G}} }}}} in Ω.

The proof will show the existence of some {{math|1={{var|ω}} ∈ Ω}} for which {{mvar|G}}<sub>{{mvar|ω}}</sub> has ''p<sup>k</sup>'' elements, providing the desired subgroup. This is the maximal possible size of a stabilizer subgroup {{mvar|G}}<sub>{{mvar|ω}}</sub>, since for any fixed element {{math|1={{var|α}} ∈ {{var|ω}} ⊆ {{var|G}}}}, the right coset {{mvar|G}}<sub>{{mvar|ω}}</sub>{{mvar|α}} is contained in {{mvar|ω}}; therefore, {{math|1={{abs|{{var|G}}<sub>{{var|ω}}</sub>}} = {{abs|{{var|G}}<sub>{{var|ω}}</sub>{{mvar|α}}}} ≤ {{abs|{{var|ω}}}} = {{var|p}}<sup>{{var|k}}</sup>}}.

By the [[Group action (mathematics)#Orbits and stabilizers|orbit-stabilizer theorem]] we have {{math|1={{abs|{{var|G}}<sub>{{var|ω}}</sub>}} {{abs|{{var|G}}{{var|ω}}}} = {{abs|{{var|G}}}}}} for each {{math|1={{var|ω}} ∈ Ω}}, and therefore using the [[additive p-adic valuation]] ''ν<sub>p</sub>'', which counts the number of factors ''p'', one has {{math|1=''ν<sub>p</sub>''({{abs|{{var|G}}<sub>{{var|ω}}</sub>}}) + ''ν<sub>p</sub>''({{abs|{{var|G}}{{var|ω}}}}) = ''ν<sub>p</sub>''({{abs|{{var|G}}}}) = ''k'' + ''r''}}. This means that for those {{mvar|ω}} with {{math|1={{abs|{{var|G}}<sub>{{var|ω}}</sub>}} = ''p<sup>k</sup>''}}, the ones we are looking for, one has {{math|1=''ν<sub>p</sub>''({{abs|{{var|G}}{{var|ω}}}}) = ''r''}}, while for any other {{mvar|ω}} one has {{math|1=''ν<sub>p</sub>''({{abs|{{var|G}}{{var|ω}}}}) > ''r''}} (as {{math|1=0 < {{abs|{{var|G}}<sub>{{var|ω}}</sub>}} < ''p<sup>k</sup>''}} implies {{math|1=''ν<sub>p</sub>''({{abs|{{var|G}}<sub>{{var|ω}}</sub>}}) < ''k'')}}. Since {{math|1={{abs|Ω}}}} is the sum of {{math|1={{abs|{{var|G}}{{var|ω}}}}}} over all distinct orbits {{mvar|G}}{{mvar|ω}}, one can show the existence of ω of the former type by showing that {{math|1=''ν<sub>p</sub>''({{abs|Ω}}) = ''r''}} (if none existed, that valuation would exceed ''r''). This is an instance of [[Kummer's theorem]] (since in base ''p'' notation the number {{math|1={{abs|{{var|G}}}}}} ends with precisely ''k'' + ''r'' digits zero, subtracting ''p<sup>k</sup>'' from it involves a carry in ''r'' places), and can also be shown by a simple computation:

:<math>|\Omega | ={p^km \choose p^k} = \prod_{j=0}^{p^k - 1} \frac{p^k m - j}{p^k - j} =  m\prod_{j=1}^{p^{k} - 1} \frac{p^{k - \nu_p(j)} m - j/p^{\nu_p(j)}}{p^{k - \nu_p(j)} - j/p^{\nu_p(j)}} </math>

and no power of ''p'' remains in any of the factors inside the product on the right. Hence {{math|1=''ν<sub>p</sub>''({{abs|Ω}}) = ''ν<sub>p</sub>''(''m'') = ''r''}}, completing the proof.

It may be noted that conversely every subgroup ''H'' of order ''p<sup>k</sup>'' gives rise to sets {{math|1={{var|ω}} ∈ Ω}} for which {{var|G}}<sub>{{var|ω}}</sub> = ''H'', namely any one of the ''m'' distinct cosets ''Hg''.}}

{{math theorem|name=Lemma|1=Let {{mvar|H}} be a finite {{mvar|p}}-group, let Ω be a finite set acted on by {{mvar|H}}, and let Ω<sub>0</sub> denote the set of points of Ω that are fixed under the action of {{mvar|H}}.  Then {{math|1={{abs|Ω}} ≡ {{abs|Ω<sub>0</sub>}} (mod&nbsp;{{var|p}})}}.}}

{{math proof|1=Any element {{math|1={{var|x}} ∈ Ω}} not fixed by {{mvar|H}} will lie in an orbit of order {{math|1={{abs|{{var|H}}}}/{{abs|''H<sub>x</sub>''}}}} (where ''H<sub>x</sub>'' denotes the [[Group action (mathematics)#Orbits and stabilizers|stabilizer]]), which is a multiple of {{mvar|p}} by assumption.  The result follows immediately by writing {{math|1={{abs|Ω}}}} as the sum of {{math|1={{abs|{{var|H}}{{var|x}}}}}} over all distinct orbits {{mvar|H}}{{mvar|x}} and reducing mod {{mvar|p}}.}}

{{math theorem|note=2|1=If ''H'' is a ''p''-subgroup of {{mvar|G}} and ''P'' is a Sylow ''p''-subgroup of {{mvar|G}}, then there exists an element ''g'' in {{mvar|G}} such that ''g''<sup>−1</sup>''Hg'' ≤ ''P''. In particular, all Sylow ''p''-subgroups of {{mvar|G}} are [[conjugacy class|conjugate]] to each other (and therefore [[isomorphism|isomorphic]]), that is, if ''H'' and ''K'' are Sylow ''p''-subgroups of {{mvar|G}}, then there exists an element ''g'' in {{mvar|G}} with ''g''<sup>−1</sup>''Hg'' = ''K''.}}

{{math proof|1=Let Ω be the set of left [[coset]]s of ''P'' in {{mvar|G}} and let ''H'' act on Ω by left multiplication. Applying the Lemma to ''H'' on Ω, we see that {{math|1={{abs|Ω<sub>0</sub>}} ≡ {{abs|Ω}} = [{{var|G}} : ''P''] (mod&nbsp;''p'')}}.  Now <math>p \nmid [G:P]</math> by definition so <math>p \nmid |\Omega_0|</math>, hence in particular {{math|1={{abs|Ω<sub>0</sub>}} ≠ 0}} so there exists some {{math|1=''gP'' ∈ Ω<sub>0</sub>}}. With this ''gP'', we have ''hgP'' = ''gP'' for all {{math|1={{var|h}} ∈ {{var|H}}}}, so ''g''<sup>−1</sup>''HgP'' = ''P'' and therefore ''g''<sup>−1</sup>''Hg'' ≤ ''P''. Furthermore, if ''H'' is a Sylow ''p''-subgroup, then {{math|1={{abs|''g''<sup>−1</sup>''Hg''}} = {{abs|''H''}} = {{abs|''P''}}}} so that ''g''<sup>−1</sup>''Hg'' = ''P''.}}

{{math theorem|note=3|1=Let ''q'' denote the order of any Sylow ''p''-subgroup ''P'' of a finite group {{var|G}}. Let ''n<sub>p</sub>'' denote the number of Sylow ''p''-subgroups of {{var|G}}.  Then (a) {{math|1=''n<sub>p</sub>'' = [{{var|G}} : ''N<sub>G</sub>''(''P'')]}} (where ''N<sub>G</sub>''(''P'') is the [[normalizer]] of ''P''), (b) {{math|1=''n<sub>p</sub>''}} divides {{math|1={{abs|{{var|G}}}}/''q''}}, and (c) {{math|1=''n<sub>p</sub>'' ≡ 1 (mod&nbsp;''p'')}}.}}

{{math proof|1=Let Ω be the set of all Sylow ''p''-subgroups of {{mvar|G}} and let {{mvar|G}} act on Ω by conjugation. Let {{math|1=''P'' ∈ Ω}} be a Sylow ''p''-subgroup. By Theorem 2, the orbit of ''P'' has size ''n<sub>p</sub>'', so by the orbit-stabilizer theorem {{math|1=''n<sub>p</sub>'' = [{{var|G}} : {{var|G}}<sub>''P''</sub>]}}. For this group action, the stabilizer {{mvar|G}}<sub>''P''</sub> is given by {{math|1={{mset|1= {{var|g}} ∈ {{var|G}} {{pipe}} ''gPg''<sup>−1</sup> = ''P'' }}  = ''N''<sub>G</sub>(''P'')}}, the normalizer of ''P'' in {{var|G}}. Thus, {{math|1=''n<sub>p</sub>'' = [{{var|G}} : ''N<sub>G</sub>''(''P'')]}}, and it follows that this number is a divisor of {{math|1=[{{var|G}} : ''P''] = {{abs|{{var|G}}}}/''q''}}.

Now let ''P'' act on Ω by conjugation, and again let Ω<sub>0</sub> denote the set of fixed points of this action. Let {{math|1=''Q'' ∈ Ω<sub>0</sub>}} and observe that then {{math|1=''Q'' = ''xQx''<sup>−1</sup>}} for all {{math|1=''x'' ∈ ''P''}} so that ''P'' ≤ ''N<sub>G</sub>''(''Q'').  By Theorem 2, ''P'' and ''Q'' are conjugate in ''N<sub>G</sub>''(''Q'') in particular, and ''Q'' is normal in ''N<sub>G</sub>''(''Q''), so then ''P'' = ''Q''.  It follows that Ω<sub>0</sub> = {''P''} so that, by the Lemma, {{math|1={{abs|Ω}} ≡ {{abs|Ω<sub>0</sub>}} = 1 (mod&nbsp;''p'')}}.}}