Hall–Petresco identity

<h2 id="statement">Statement</h2>
The Hall–Petresco identity states that if x and y are elements of a group G and m is a positive integer then

x
          
            m
          
        
        
          y
          
            m
          
        
        =
        (
        x
        y
        
          )
          
            m
          
        
        
          c
          
            2
          
          
            
              
                (
              
              
                m
                2
              
              
                )
              
            
          
        
        
          c
          
            3
          
          
            
              
                (
              
              
                m
                3
              
              
                )
              
            
          
        
        ⋯
        
          c
          
            m
            −
            1
          
          
            
              
                (
              
              
                m
                
                  m
                  −
                  1
                
              
              
                )
              
            
          
        
        
          c
          
            m
          
        
      
    
    {\displaystyle x^{m}y^{m}=(xy)^{m}c_{2}^{\binom {m}{2}}c_{3}^{\binom {m}{3}}\cdots c_{m-1}^{\binom {m}{m-1}}c_{m}}

where each ci is in the subgroup Ki of the descending central series of G.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Baker%25E2%2580%2593Campbell%25E2%2580%2593Hausdorff_formula/k3HtPR2N">Baker–Campbell–Hausdorff formula</a></li>
<li><a href="/facts/Universal_enveloping_algebra/VJ3Syz4W">Algebra of symbols</a></li></ul>

<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/Philip_Hall/4Gal202l">Hall, Philip</a> (1934), "A contribution to the theory of groups of prime-power order", Proceedings of the London Mathematical Society, 36: 29–95, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fplms%2Fs2-36.1.29">10.1112/plms/s2-36.1.29</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>
<li>Petresco, Julian (1954), "Sur les commutateurs", Mathematische Zeitschrift, 61 (1): 348–356, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01181351">10.1007/BF01181351</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0066380">0066380</a></li></ul>

Hall–Petresco identity open-in-new

Hall–Petresco identity