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For a group  G , letN  ⁢   (  G  )${N(G)}$    be the set of lengths of conjugacy classes of  G . In this article, we prove a conjecture of J. G. Thompson for the finite simple groupsB  n   ⁢   (  q  )${B_{n}(q)}$    andC  n   ⁢   (  q  )${C_{n}(q)}$   , i.e., we show that every finite group  G  with the propertyZ  ⁢   (  G  )    =  1${Z(G)=1}$    andN  ⁢   (  G  )    =   N  ⁢   (  S  )${N(G)=N(S)}$    is necessarily isomorphic to  S , whereS  ∈   {    B  n   ⁢   (  q  )    ,    C  n   ⁢   (  q  )    }${S\in\{B_{n}(q),C_{n}(q)\}}$   . As a consequence, we show that ifZ  ⁢   (  G  )    ∩   Φ  ⁢   (  G  )=  1${Z(G)\cap\Phi(G)=1}$    andN  ⁢   (  G  )    =   N  ⁢   (  S  )${N(G)=N(S)}$   , then  G  is isomorphic toZ  ⁢   (  G  )    ×  S${Z(G)\times S}$   .

Thompson’s conjecture for finite simple groups of Lie type B n ${B_{n}}$ and C n ${C_{n}}$