# Groups of exponent 4

Is there a classification of finite nonabelian 2-groups of exponent 4?

What about, finite nonabelian 3-groups of exponent 3?

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There is no classification of finite groups of exponent 4. You might find this paper interesting - it contains lots of information about how the group Burnside group $B(m,4)$ grows (all $m$-generator exponent-4 groups are quotients of this group).

There is also no classification of finite groups of exponent 3. However it is known that these groups must be $2$-Engel and class three. Furthermore in this case the precise size of the corresponding Burnside group is known: $B(m,3)$ is a finite group of size $3^{m + \binom{m}{2} + \binom{m}{3}}$.

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What about, finite nonabelian 3-groups of exponent 3?

Those are all quotients of the Burnside group $B(m,3)$ for some value for $m$.

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Tom, you beat me by 1 minute!! –  Nick Gill Nov 14 '12 at 8:52
But your answer is more complete than mine :) –  Tom De Medts Nov 14 '12 at 12:55