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Let  G  be a compact Hausdorff group. We show that if for every elementthere exists a positive integersuch that  x qis Engel, then  G  is locally virtually nilpotent. Furthermore, we show that if  G  is a finitely generated compact Hausdorff group in which any commutatorin  G  is Engel, then the commutator subgroupis locally nilpotent.

On compact groups with Engel-like conditions