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We say that an element    g    of a group    G    is almost right Engel if there is a finite setR   (  g  )such that for everyx  ∈  G    , there is a positive integern  (  x  ,  g  )such that[  …  [  [  g  ,  x  ]  ,  x  ]  ,  …  ,  x  ]  ∈  R   (  g  )if    x    is repeated at leastn  (  x  ,  g  )times. Thus,    g    is a right Engel element precisely when we can chooseR   (  g  )   =   {  1  }. We prove that if all elements of a compact (Hausdorff) group    G    are almost right Engel, then    G    has a finite normal subgroup    N    such thatG  /  Nis locally nilpotent. If in addition there is a uniform bound∣  R   (  g  )   ∣  ⩽  mfor the orders of the corresponding sets, then the subgroup    N    can be chosen of order bounded in terms of    m   . The proofs use the Wilson–Zelmanov theorem saying that profinite Engel groups are locally nilpotent and previous results of the authors about compact groups in which all elements are almost left Engel.

Compact groups in which all elements are almost right Engel