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?

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 $3^{m + \binom{m}{2} + \binom{m}{3}}$.

• Please activate the link of "this paper". Many thanks. – user100452 Oct 30 '16 at 10:19
• Access was denied on the link: was it a paper of M. Vaughn-Lee, who has worked a lot on such groups? – Geoff Robinson Oct 30 '16 at 11:41
• @user100452, Dropbox seems to have changed its policy on link sharing, and I can't figure out how to make the paper public. Please email me -- my details are on my user page -- and I'll happily send you a copy of the paper. – Nick Gill Oct 31 '16 at 9:02
• @GeoffRobinson, It's a paper by Avinoam Mann: On the orders of groups of exponent 4. – Nick Gill Oct 31 '16 at 9:04
• @NickGill : Thanks.. Vaughn-Lee has done a lot of work on groups of exponent $4$. – Geoff Robinson Oct 31 '16 at 9:47

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

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

• 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