Splitting automorphisms of free Burnside groups. (Q2842987)

An automorphism \(f\) of a group \(G\) is called a splitting automorphism of period \(n\) if \(f^n=1\) and \(xx^fx^{f^2}\cdots x^{f^{n-1}}=1\) for all \(x\in G\). This is equivalent to all elements in the semidirect product \(G\langle f\rangle\) outside \(G\) having order dividing \(n\). It is proved that if \(f\) is a splitting automorphism of odd period \(n\geq 1009\) of a free Burnside group \(B(m,n)\) of exponent \(n\) and the order of \(f\) is a prime, then \(f\) is an inner automorphism. The proof relies on the author's results [in Izv. Math. 75, No. 2, 223-237 (2011); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 75, No. 2, 3-18 (2011; Zbl 1227.20030)]. Similar results for very large prime \(n\) were also obtained by \textit{E. A. Cherepanov} [Int. J. Algebra Comput. 16, No. 5, 839-847 (2006; Zbl 1115.20024)].