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Now we claim that the Restricted Burnside Problem has an
affirmative answer for exponent *p*^{n}. Let us examine this claim.
To prove the Restricted Burnside Problem we need to show that the
order *p*^{a} of a *k*-generated *p*-group *G* of exponent *p*^{n} is uniformly
bounded by some constant
*p*^{a(k, n)}. Now we have
dim_{Z/pZ}*L*(*G*) = *a*. Thus it is enough to bound the
dimension of the Lie algebra *L*(*G*). As in the case of the free group
we can construct a *universal* Lie algebra *L* which is generated
by *k* elements and satisfies the Higman and Sanov identities.
Assuming the above theorem *L* is nilpotent. But then the abelian
sub-quotients of the central series of *L* have a specified number of
generators in terms of the generators of *L* and are thus finite
dimensional. Thus *L* is itself finite dimensional, say of dimension
*a*(*k*, *n*). Since any *L*(*G*) is a quotient of *L* its dimension is also
bounded by *a*(*k*, *n*) and this proves the result.
The rest of the Restricted Burnside Problem now follows since we have
the result of Hall and Higman and also a complete Classification of
Finite Simple groups by Feit, Thompson, Aschbacher et al.

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Kapil Hari Paranjape
2002-11-22