Abstract
We prove that a locally graded group whose proper subgroups are Engel (respectively, [Formula: see text]-Engel) is either Engel (respectively, [Formula: see text]-Engel) or finite. We also prove that a group of infinite rank whose proper subgroups of infinite rank are Engel (respectively, [Formula: see text]-Engel) is itself Engel (respectively, [Formula: see text]-Engel), provided that [Formula: see text] belongs to the Černikov class [Formula: see text], which is the closure of the class of periodic locally graded groups by the closure operations Ṕ, P̀, R and L.
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