ANALOGUES OF NIELSEN’S AND MAGNUS’S THEOREMS FOR FREE BURNSIDE GROUPS OF PERIOD 3

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Submitted: 2025-02-04; Published: 2025-02-04

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Abstract

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 Fm of rank m. We show that the kernel of the natural homomorphism Aut(B(2,3))→GL2(Z3) is the group of inner automorphisms of B(2,3).