ANALOGUES OF NIELSEN'S AND MAGNUS'S THEOREMS FOR FREE BURNSIDE GROUPS OF PERIOD 3 - Atabekyan, V.S.; Aslanyan, H.T.; Grigoryan, H.A.; Grigoryan, A.E. (Proceedings of the YSU A: Physical and Mathematical Sciences, 2017)
We prove that the free Burnside groups $B(m,3)$ of period 3 and rank $m≥1$ have Magnus's property, that is if in $B(m,3)$ the normal closures of $r$ and $s$ coincide, then $r$ is conjugate to $s$ or $s^{-1}$. We also prove that any automorphism of $B(m,3)$ induced by a Nielsen automorphism of the free group $F_m$ of rank $m$. We show that the kernel of the natural homomorphism $\text{Aut}(B(2,3)) → GL_2(\mathbb{Z}_3)$ is the group of inner automorphisms of $B(2,3)$.
Keywords: Nielsen automorphism, Magnus’s property, tame automorphism, free Burnside group, free group