Burnside groups : a few questions

by Olivier Bégassat   Last Updated July 03, 2018 20:20 PM

Let $G=B(n,e)=F_n/\langle\langle F_n^e\rangle\rangle$ be the Burnside group on $n$ generators with exponent $e$, i.e. the quotient of the free group on $n$ generators $F_n$ by the normal subgroup generated by the set of $e$-th powers $F_n^e=\{x^e\mid x\in F_n\}$.

It is a theorem that for odd $e>664$, $G$ is infinite. Here are my questions :

1. Does $G$ have a trivial center?
2. Do the automorphisms of $G$ act transitively on $G^*=G-\{1\}$? What about the inner automorphisms?
3. If the action is indeed transitive, how transitive is it? I.e. are there simple criteria for determining whether a $k$-tuple $(g_1,\dots,g_k)$ can be transformed to $(h_1,\dots,h_k)$ via some automorphism?
4. Suppose $e$ is prime and $g,h$ don't commute: is the subgroup $\langle g,h\rangle$ isomorphic to $B(2,e)$?

References welcome.

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