4 The Proof of the Problem

Now we claim that the Restricted Burnside Problem has an affirmative answer for exponent pⁿ. 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 pⁿ is uniformly bounded by some constant p^{a(k, n)}. Now we have dim_{$\Bbb$Z/p$\Bbb$Z}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.