A local approach to stability groups (Q6618165)

Let \(G\) be a group and let \(G=G_{0}> G_{1} > \dots >G_{m}=1\) be a chain of subgroups of \(G\). The stability group \(\mathcal{A}\) of the chain is the set of all automorphisms \(\alpha \in \mathrm{Aut}(G)\) such that \(\big (g_{i}G_{i+1}\big)^{\alpha}= g_{i}G_{i+1}\) for all \(g_{i} \in G_{i}\) and for each \(i \in \{0, 1, \dots , m -1\}\). A nice result by \textit{P. Hall} [Ill. J. Math. 2, No. 4B, 787--801 (1958; Zbl 0084.25602)], states that \(\mathcal{A}\) is nilpotent of class at most \(\binom{m}{2}\).\N\NIn the paper under review, the authors, using only elementary commutator calculus, prove the following result: Let \(G\) be a group and let \(n\in \mathbb{N}\). If \(y \in G\) and \(X \leq G\) satisfy \([y, x_{1},\dots, x_{n}]=1=[y^{-1}, x_{1}, \dots , x_{n}]\) for all \(x_{1}, \dots, x_{n} \in X\), then \([\gamma_{f(n)}(X),y]=1\), where \(f(n)=\binom{n}{2}+1\).\N\NAn example constructed by Hall [loc. cit.] shows that the bound \(f(3)=4\) is actually attained.