Automorphism tower problem and semigroup of endomorphisms for free Burnside groups

TL;DR

This paper proves that the automorphism group of free Burnside groups with large odd exponents is complete, solves the automorphism tower problem for these groups, and characterizes automorphisms of their endomorphism semigroups.

Contribution

It establishes the completeness of automorphism groups of free Burnside groups for large odd exponents and solves the automorphism tower problem for these groups.

Findings

Automorphism group Aut(B(m,n)) is complete for odd n≥1003.

The automorphism tower of B(m,n) is as short as that of free groups.

Every automorphism of End(B(m,n)) is a conjugation by an element of Aut(B(m,n)).

Abstract

We have proved that the group of all inner automorphisms of the free Burnside group $B(m,n)$ is the unique normal subgroup in $Aut(B(m,n))$ among all its subgroups, which are isomorphic to free Burnside group $B(s,n)$ of some rank $s$ for all odd $n\ge1003$ and $m>1$. It follows that the group of automorphisms $Aut(B(m,n))$ of the free Burnside group $B(m,n)$ is complete for odd $n\ge1003$, that is it has a trivial center and any automorphism of $Aut(B(m,n))$ is inner. Thus, for groups $B(m,n)$ is solved the automorphism tower problem and is showed that it is as short as the automorphism tower of the absolutely free groups. Moreover, proved that every automorphism of $End(B(m,n))$ is a conjugation by an element of $Aut(B(m,n))$.