  
  [1m[4m[31m4. Examples[0m
  
  
  [1m[4m[31m4.1 Right Engel elements[0m
  
  An  old problem in the context of Engel elements is the question: Is a right
  n-Engel element left n-Engel? It is known that the answer is no. For details
  about the history of the problem, see [MW94]. In this paper the authors show
  that for n>4 there are nilpotent groups with right n-Engel elements no power
  of  which  is  a left n-Engel element. The insight was based on computations
  with the ANU NQ which we reproduce here. We also show the cases 5>n.
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35mgap> RequirePackage( "nq" );[0m
    [22m[35mtrue[0m
    [22m[35mgap> ##  SetInfoLevel( InfoNQ, 1 );[0m
    [22m[35mgap> ##[0m
    [22m[35mgap> ##  setup calculation[0m
    [22m[35mgap> ##[0m
    [22m[35mgap> et := ExpressionTrees( "a", "b", "x" );[0m
    [22m[35m[ a, b, x ][0m
    [22m[35mgap> a := et[1];; b := et[2];; x := et[3];;[0m
    [22m[35mgap> [0m
    [22m[35mgap> ##[0m
    [22m[35mgap> ##  define the group for n = 2,3,4,5[0m
    [22m[35mgap> ##[0m
    [22m[35mgap> [0m
    [22m[35mgap> rengel := LeftNormedComm( [a,x,x] );[0m
    [22m[35mComm( a, x, x )[0m
    [22m[35mgap> G := rec( generators := et, relations := [rengel] );[0m
    [22m[35mrec( generators := [ a, b, x ], relations := [ Comm( a, x, x ) ] )[0m
    [22m[35mgap> ## The following is equivalent to:[0m
    [22m[35mgap> ##   NilpotentQuotient( : input_string := NqStringExpTrees( G, [x] ) )[0m
    [22m[35mgap> H := NilpotentQuotient( G, [x] );[0m
    [22m[35mPcp-group with orders [ 0, 0, 0 ][0m
    [22m[35mgap> LeftNormedComm( [ H.2,H.1,H.1 ] );[0m
    [22m[35mid[0m
    [22m[35mgap> LeftNormedComm( [ H.1,H.2,H.2 ] );[0m
    [22m[35mid[0m
  [22m[35m------------------------------------------------------------------[0m
  
  This shows that each right 2-Engel element in a finitely generated nilpotent
  group  is  a  left 2-Engel element. Note that the group above is the largest
  nilpotent  group  generated  by two elements, one of which is right 2-Engel.
  Every  nilpotent group generated by an arbitrary element and a right 2-Engel
  element is a homomorphic image of the group H.
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35mgap> rengel := LeftNormedComm( [a,x,x,x] );[0m
    [22m[35mComm( a, x, x, x )[0m
    [22m[35mgap> G := rec( generators := et, relations := [rengel] );[0m
    [22m[35mrec( generators := [ a, b, x ], relations := [ Comm( a, x, x, x ) ] )[0m
    [22m[35mgap> H := NilpotentQuotient( G, [x] );[0m
    [22m[35mPcp-group with orders [ 0, 0, 0, 0, 0, 4, 2, 2 ][0m
    [22m[35mgap> LeftNormedComm( [ H.1,H.2,H.2,H.2 ] );[0m
    [22m[35mid[0m
    [22m[35mgap> h := LeftNormedComm( [ H.2,H.1,H.1,H.1 ] );[0m
    [22m[35mg6^2*g7*g8[0m
    [22m[35mgap> Order( h );[0m
    [22m[35m4[0m
  [22m[35m------------------------------------------------------------------[0m
  
  The  element  h  has order 4. In a nilpotent group without 2-torsion a right
  3-Engel element is left 3-Engel.
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35mgap> rengel := LeftNormedComm( [a,x,x,x,x] );[0m
    [22m[35mComm( a, x, x, x, x )[0m
    [22m[35mgap> G := rec( generators := et, relations := [rengel] );[0m
    [22m[35mrec( generators := [ a, b, x ], relations := [ Comm( a, x, x, x, x ) ] )[0m
    [22m[35mgap> H := NilpotentQuotient( G, [x] );[0m
    [22m[35mPcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 12, 0, 5, 10, 2, 0, 30, [0m
    [22m[35m  5, 2, 5, 5, 5, 5 ][0m
    [22m[35mgap> LeftNormedComm( [ H.1,H.2,H.2,H.2,H.2 ] );[0m
    [22m[35mid[0m
    [22m[35mgap> h := LeftNormedComm( [ H.2,H.1,H.1,H.1,H.1 ] );[0m
    [22m[35mg9*g10^2*g11^10*g12^5*g13^2*g14^8*g15*g16^6*g17^10*g18*g20^4*g21^4*g22^2*g23^2[0m
    [22m[35mgap> Order( h );[0m
    [22m[35m60[0m
  [22m[35m------------------------------------------------------------------[0m
  
  The   previous   calculation   shows  that  in  a  nilpotent  group  without
  2,3,5-torsion a right 4-Engel element is left 4-Engel.
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35mgap> rengel := LeftNormedComm( [a,x,x,x,x,x] );[0m
    [22m[35mComm( a, x, x, x, x, x )[0m
    [22m[35mgap> G := rec( generators := et, relations := [rengel] );[0m
    [22m[35mrec( generators := [ a, b, x ], relations := [ Comm( a, x, x, x, x, x ) ] )[0m
    [22m[35mgap> H := NilpotentQuotient( G, [x], 9 );[0m
    [22m[35mPcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 30, [0m
    [22m[35m  0, 0, 30, 0, 3, 6, 0, 0, 10, 30, 0, 0, 0, 0, 30, 30, 0, 0, 3, 6, 5, 2, 0, [0m
    [22m[35m  2, 408, 2, 0, 0, 0, 10, 10, 30, 10, 0, 0, 0, 3, 3, 3, 2, 204, 6, 6, 0, 10, [0m
    [22m[35m  10, 10, 2, 2, 2, 0, 300, 0, 0, 18 ][0m
    [22m[35mgap> LeftNormedComm( [ H.1,H.2,H.2,H.2,H.2,H.2 ] );[0m
    [22m[35mid[0m
    [22m[35mgap> h := LeftNormedComm( [ H.2,H.1,H.1,H.1,H.1,H.1 ] );;[0m
    [22m[35mgap> Order( h );[0m
    [22m[35minfinity[0m
  [22m[35m------------------------------------------------------------------[0m
  
  Finally,  we  see  that in a torsion-free group a right 5-Engel element need
  not be a left 5-Engel element.
  
