# Burnside problem

\begin{conjecture} If a group has $r$ generators and exponent $n$, is it necessarily finite? \end{conjecture}

It is possible to define the $\emph{free Burnside group}$ $B(r,n)$ to be the group generated by $x_{1}, \ldots, x_{r}$ with relations $w^{n}=1$ where $w$ ranges over every word in the generators. There is a universality property: Any homomorphism $ \phi : G \to H$ where $H$ has r generators and exponent dividing $n$ can be written as a composition of a homomorphism $ \psi : G \to B(r,n) $ with a homomorphism $ \pi : B(r,n) \to H$. Some cases of this are known: $B(1,n)$ is a cyclic group of order $n$, for any positive integer $n$. $B(r,1)$ is trivial for any positive integer $r$. $B(r,2)$ is isomorphic to the Cartesian product of $r$ cyclic groups of order $2$, for any positive integer $r$. This is because the relations make it easy to prove that the generators commute. $B(r,3)$ is a finite group, and its order is \[ 3^{r + \binom{r}{2} + \binom{r}{3}}. \] $B(r,6)$ is a finite group, and its order is \[ 2^{1 + (r-1)3^{r + \binom{r}{2} + \binom{r}{3}} }3^{1 + (r-1)2^{r + \binom{r}{2} + \binom{r}{3}} }. \] $B(r,4)$ is a finite group for any positive integer $r$. The order is known for $r$ up to $5$: \[ |B(1,4)| = 2^{2} \] \[ |B(2,4)| = 2^{12} \] \[ |B(3,4)| = 2^{69} \] \[ |B(4,4)| = 2^{422} \] \[ |B(5,4)| = 2^{2728} \] $B(r,n)$ is known to be infinite for sufficiently large $r$ and odd $n \geq 665$, as well as $r > 1$ and $n \geq 10^{48}$ divisible by $2^{9}$.

\Def {Burnside_problem} \[ % I don't know how to do line breaks in LaTeX \] \href[Burnside Problem -- from Wolfram MathWorld]{http://mathworld.wolfram.com/BurnsideProblem.html} % You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/}

## Bibliography

% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

* indicates original appearance(s) of problem.