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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:
[1] Novikov P. S., Izv. Acad. Nauk SSSR, T. 32
[2] Adian S. I., The Burnside Problem and Identities in Groups (1975)
[3] Yu Ol’shanskii A., The Geometry of Defining Relations in Groups (1989)
[4] DOI: 10.1007/978-1-4613-0235-3_12 · doi:10.1007/978-1-4613-0235-3_12
[5] DOI: 10.1016/0021-8693(80)90208-2 · Zbl 0432.20024 · doi:10.1016/0021-8693(80)90208-2
[6] DOI: 10.1016/0021-8693(80)90086-1 · Zbl 0432.20025 · doi:10.1016/0021-8693(80)90086-1
[7] DOI: 10.1016/0021-8693(80)90132-5 · Zbl 0435.20015 · doi:10.1016/0021-8693(80)90132-5
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