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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 $B(m,3)?$

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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.

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    $\begingroup$ what is that notation in the final product? $\endgroup$
    – mathlander
    Commented Oct 27, 2022 at 0:48
  • $\begingroup$ @mathlander What do you mean? $\endgroup$ Commented Oct 27, 2022 at 0:49
  • $\begingroup$ I mean the one that looks like an associator. $\endgroup$
    – mathlander
    Commented Oct 27, 2022 at 0:50
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    $\begingroup$ @matlander Also standard notation when working with varieties and nilpotent groups. $[a,b,c]=[[a,b],c]$, when following left-normed convention (which goes with $[a,b]=a^{-1}b^{-1}ab$). $\endgroup$ Commented Oct 27, 2022 at 0:56
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    $\begingroup$ @mathlander And the theorems quoted do not say "finite" anywhere. I am not in the habit of eliding finiteness. $\endgroup$ Commented Oct 29, 2022 at 15:56

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