What are all the finite groups where every nontrivial element has order $3?$

Question

I know that the powers of any group where all elements have order $3$ also has this property. I also know that the Heisenberg group has this property and that all finitely generated such groups are finite. What are all the finite groups where every nontrivial element has order $3?$ In particular, what is the order and structure of the free Burnside group $B(m,3)?$

Answer 1

Levi and van der Waerden proved groups of exponent $3$ are nilpotent of class at most $3.$ By work of Levi, they are also $2-$Engel. The complete description appears in Marshall Hall, Jr.'s The Theory of Groups, Section 18.2.

The order of $B(3,r)$ (the free Burnside group of exponent $3$ and rank $r$, following the notation of Hall) is $3^{m(r)}$, where $m(r) = r+\binom{r}{2} + \binom{r}{3}$ (Theorem 18.2.1), and every element can be written uniquely as $$g=x_1^{a_1}\cdots x_r^{a_r}\prod_{1\leq i\lt j\leq r}[x_i,x_j]^{b_{ij}} \prod_{1\leq i\lt j\lt k\leq r}[x_i,x_j,x_k]^{c_{ijk}}$$ with exponents taken modulo $3$, where the $x_i$ are the free generators; the product is given by the standard commutator collection formulas.