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Normal automorphisms of free Burnside groups of large odd exponents. (English) Zbl 1115.20024
An automorphism $$\varphi$$ of a group $$G$$ is said to be normal if every normal subgroup of $$G$$ is $$\varphi$$-invariant. The author proves that every normal automorphism of a free Burnside group $$B(m,n)$$ on $$m\geq 2$$ generators of sufficiently large odd exponent $$n$$ is inner. Earlier similar results were known for absolutely free groups (Lubotzky), free profinite groups (Jarden), and free soluble pro-$$p$$-groups (Romanovskiĭ). Moreover, the author proves that for every outer automorphism $$\varphi$$ of $$B(m,n)$$ there is a normal subgroup $$N\leq B(m,n)$$ such that $$\varphi(N)\neq N$$ and $$B(m,n)/N$$ is non-Abelian simple. It is also proved that $$B(m,n)$$ can be a normal subgroup of a group of exponent $$n$$ only as a direct factor. The proofs rely on Ol’shanskiĭ’s technique of the geometrical approach to defining relations in groups and on certain lemmas in [A. Yu. Ol’shanskiĭ, Groups, rings, Lie and Hopf algebras. Math. Appl., Dordr. 555, 179-187 (2003; Zbl 1045.20028)].

##### MSC:
 20E36 Automorphisms of infinite groups 20F50 Periodic groups; locally finite groups 20F28 Automorphism groups of groups 20F05 Generators, relations, and presentations of groups 20F06 Cancellation theory of groups; application of van Kampen diagrams
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##### References:
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