Finite groups G = [G, g] have order bounded in terms of the right or left Engel sink of g.
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In finite π-soluble groups, the rank of [G,A] for a π-group A of automorphisms is bounded in terms of r whenever every subset of commutators generates an r-generated subgroup.
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On finite groups containing an element whose Engel sink is small
Finite groups G = [G, g] have order bounded in terms of the right or left Engel sink of g.
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Local--global generation property of commutators in finite $\pi$-soluble groups
In finite π-soluble groups, the rank of [G,A] for a π-group A of automorphisms is bounded in terms of r whenever every subset of commutators generates an r-generated subgroup.