Splitting automorphisms of free Burnside groups of orders \(p^k\) are inner
V. Atabekian
An automorphism \( \varphi \) of \( G \) is called a splitting
automorphism of period \( n \), if \( \varphi ^ n = 1 \) and
\( g \, g ^ {\varphi} g ^ {\varphi ^ 2} \cdot \cdot \cdot g ^ {\varphi ^ {n-1}}\)\(= 1\)
for any element \( g \in G \). By a theorem of O. Kegel [1],
any finite group having a nontrivial splitting automorphism of prime
period is nilpotent. E. Khukhro [2] proved that any solvable
group having a nontrivial splitting automorphism of prime period
also is nilpotent.
It is easy to verify that each inner automorphism of the periodic
group of period \( n \) is its splitting automorphism of period \( n \).
However, the converse is not true.
In the Kourovka notebook, S.V. Ivanov posed the question: Let \( n \)
be a sufficiently large odd number and \( m> 1 \). Is it true that any
splitting automorphism of period \( n \) of the free Burnside group \( B (m, n) \) is inner [3, question 11.36.b]?
In the paper [4], we proved that if the order of the
automorphism \( \varphi \) of the free Burnside group \( B (m, n) \) is
prime then \( \varphi \) is an inner automorphism. In [4], it was also
proved that if \(\varphi\) is an splitting automorphism of period
\(n\) of \(B(m,n)\) then the stabilizer of any normal subgroup \(N\in \mathcal{M}_n\) under the action of \(\langle\varphi\rangle\) is not
trivial for any odd \(n\ge1003\), where \(\mathcal{M}_n\) is a specially
chosen set of normal subgroups. This statement has proved useful for
solving the mentioned problem for automorphisms of prime power orders.
We proved the following result.
Theorem.Let \( \varphi \) be a splitting automorphism of
period \( n \) of the free Burnside group \( B (m, n) \), where \( n \ge1003 \) is an arbitrary odd number. Then, if the order of \( \varphi \) is a prime power, then \( \varphi \) is an
inner automorphism.
References
O. H. Kegel, Die Nilpotenz der \(H_p\)-Gruppen, Math. Z., 75 (1961), 373–376.
E. I. Khukhro, Nilpotency of solvable groups admitting a splitting automorphism of prime order,
Algebra and Logic, 19, N1 (1980), 77–84.
The Kourovka Notebook. 11th ed., Novosibirsk (1990).
V. S. Atabekyan, Splitting automorphisms of free Burnside
groups, Mat. Sb., 204, N2 (2013), 31–38.
