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Let  V  be a finite-dimensional  FG -module, where  F  is a field of prime characteristic  p  and  G  is a group. We show that, when  r  is not a power of  p , the Lie power  L   r  ( V ) has a direct summand  B   r  ( V ), which is a direct summand of the tensor power  V  ⊗  r   such that dim  B   r   ( V )/ dim  L   r   ( V ) → 1 as  r →∞. Similarly, for the same values of  r , we obtain a projective submodule  C ( r ) of the Lie module Lie ( r ) over  F  such that dim  C ( r )/dim Lie ( r )→1 as  r →∞.

ASYMPTOTIC BEHAVIOUR OF LIE POWERS AND LIE MODULES