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We prove that if $A$ is a $\\pi$-group of automorphisms of a $\\pi$-soluble finite group $G$ such that any subset of $I_G(A)$ generates a subgroup that can be generated by $r$ elements, then the rank of $[G,A]$ is bounded in terms of $r$. Examples show that such a result does not hold without the assumption of $\\pi$-solubility. 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Guralnick","submitted_at":"2025-05-05T20:37:51Z","abstract_excerpt":"For a group $A$ acting by automorphisms on a group $G$, let $I_G(A)$ denote the set of commutators $[g,a]=g^{-1}g^a$, where $g\\in G$ and $a\\in A$, so that $[G,A]$ is the subgroup generated by $I_G(A)$. We prove that if $A$ is a $\\pi$-group of automorphisms of a $\\pi$-soluble finite group $G$ such that any subset of $I_G(A)$ generates a subgroup that can be generated by $r$ elements, then the rank of $[G,A]$ is bounded in terms of $r$. Examples show that such a result does not hold without the assumption of $\\pi$-solubility. 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