Commutator collecting process

<h2 id="statement">Statement</h2>
The commutator collecting process is usually stated for <a href="/facts/Free_group/zPo2NIGb">free groups</a>, as a similar theorem then holds for any group by writing it as a <a href="/facts/Quotient_group/HlwpHtjb">quotient</a> of a free group. 
Suppose F1 is a free group on generators a1, ..., am. Define the descending <a href="/facts/Central_series/Hw9lbKGf">central series</a> by putting

Fn+1 = [Fn, F1]
The basic commutators are elements of F1 defined and ordered as follows:

<ul><li>The basic commutators of weight 1 are the <a href="/facts/Group_generators/B9q5lnFY">generators</a> a1, ..., am.</li>
<li>The basic commutators of weight w > 1 are the elements [x, y] where x and y are basic commutators whose weights sum to w, such that x > y and if x = [u, v] for basic commutators u and v then v ≤ y.</li></ul>
Commutators are ordered so that x > y if x has weight greater than that of y, and for commutators of any fixed weight some total ordering is chosen. 
Then Fn /Fn+1 is a <a href="/facts/Finitely_generated_abelian_group/dwtl7mp4">finitely generated free abelian group</a> with a basis consisting of basic commutators of weight n.
Then any element of F can be written as

g
        =
        
          c
          
            1
          
          
            
              n
              
                1
              
            
          
        
        
          c
          
            2
          
          
            
              n
              
                2
              
            
          
        
        ⋯
        
          c
          
            k
          
          
            
              n
              
                k
              
            
          
        
        c
      
    
    {\displaystyle g=c_{1}^{n_{1}}c_{2}^{n_{2}}\cdots c_{k}^{n_{k}}c}

where the ci are the basic commutators of weight at most m arranged in order, and c is a product of commutators of weight greater than m, and the ni are <a href="/facts/Integer/PUwwrawx">integers</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Hall%E2%80%93Petresco_identity/mLge2sUd">Hall–Petresco identity</a></li>
<li><a href="/facts/Monoid_factorisation/XeHmleRj">Monoid factorisation</a></li></ul>

<h2 id="reading">Reading</h2>
<ul><li><a href="/facts/Marshall_Hall_(mathematician)/wLpjsYnL">Hall, Marshall</a> (1959), The theory of groups, Macmillan, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0103215">0103215</a></li>
<li><a href="/facts/Bertram_Huppert/eEL3KWHQ">Huppert, B.</a> (1967), Endliche Gruppen (in German), Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, pp. 90–93, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-03825-2, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0224703">0224703</a>, <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/527050">527050</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Hall, Philip (1934), "A contribution to the theory of groups of prime-power order", Proceedings of the London Mathematical Society, 36: 29–95, doi:10.1112/plms/s2-36.1.29 <a href="/wiki/Philip_Hall" target="_blank">/wiki/Philip_Hall</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">W. Magnus (1937), "Über Beziehungen zwischen höheren Kommutatoren", J. Grelle 177, 105-115. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Commutator collecting process open-in-new

Commutator collecting process