  
  [1X4 [33X[0;0YExamples[133X[101X
  
  
  [1X4.1 [33X[0;0YRight Engel elements[133X[101X
  
  [33X[0;0YAn  old problem in the context of Engel elements is the question: Is a right
  [22Xn[122X-Engel element left [22Xn[122X-Engel? It is known that the answer is no. For details
  about the history of the problem, see [NN94]. In this paper the authors show
  that for [22Xn>4[122X there are nilpotent groups with right [22Xn[122X-Engel elements no power
  of  which  is  a left [22Xn[122X-Engel element. The insight was based on computations
  with the ANU NQ which we reproduce here. We also show the cases [22X5>n[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "nq" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27X##  SetInfoLevel( InfoNQ, 1 );[127X[104X
    [4X[25Xgap>[125X [27X##[127X[104X
    [4X[25Xgap>[125X [27X##  setup calculation[127X[104X
    [4X[25Xgap>[125X [27X##[127X[104X
    [4X[25Xgap>[125X [27Xet := ExpressionTrees( "a", "b", "x" );[127X[104X
    [4X[28X[ a, b, x ][128X[104X
    [4X[25Xgap>[125X [27Xa := et[1];; b := et[2];; x := et[3];;[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27X##[127X[104X
    [4X[25Xgap>[125X [27X##  define the group for n = 2,3,4,5[127X[104X
    [4X[25Xgap>[125X [27X##[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27Xrengel := LeftNormedComm( [a,x,x] );[127X[104X
    [4X[28XComm( a, x, x )[128X[104X
    [4X[25Xgap>[125X [27XG := rec( generators := et, relations := [rengel] );[127X[104X
    [4X[28Xrec( generators := [ a, b, x ], relations := [ Comm( a, x, x ) ] )[128X[104X
    [4X[25Xgap>[125X [27X## The following is equivalent to:[127X[104X
    [4X[25Xgap>[125X [27X##   NilpotentQuotient( : input_string := NqStringExpTrees( G, [x] ) )[127X[104X
    [4X[25Xgap>[125X [27XH := NilpotentQuotient( G, [x] );[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XLeftNormedComm( [ H.2,H.1,H.1 ] );[127X[104X
    [4X[28Xid[128X[104X
    [4X[25Xgap>[125X [27XLeftNormedComm( [ H.1,H.2,H.2 ] );[127X[104X
    [4X[28Xid[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis 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 [22XH[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrengel := LeftNormedComm( [a,x,x,x] );[127X[104X
    [4X[28XComm( a, x, x, x )[128X[104X
    [4X[25Xgap>[125X [27XG := rec( generators := et, relations := [rengel] );[127X[104X
    [4X[28Xrec( generators := [ a, b, x ], relations := [ Comm( a, x, x, x ) ] )[128X[104X
    [4X[25Xgap>[125X [27XH := NilpotentQuotient( G, [x] );[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0, 0, 0, 4, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XLeftNormedComm( [ H.1,H.2,H.2,H.2 ] );[127X[104X
    [4X[28Xid[128X[104X
    [4X[25Xgap>[125X [27Xh := LeftNormedComm( [ H.2,H.1,H.1,H.1 ] );[127X[104X
    [4X[28Xg6^2*g7*g8[128X[104X
    [4X[25Xgap>[125X [27XOrder( h );[127X[104X
    [4X[28X4[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  element  [22Xh[122X  has order [22X4[122X. In a nilpotent group without [22X2[122X-torsion a right
  3-Engel element is left 3-Engel.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrengel := LeftNormedComm( [a,x,x,x,x] );[127X[104X
    [4X[28XComm( a, x, x, x, x )[128X[104X
    [4X[25Xgap>[125X [27XG := rec( generators := et, relations := [rengel] );[127X[104X
    [4X[28Xrec( generators := [ a, b, x ], relations := [ Comm( a, x, x, x, x ) ] )[128X[104X
    [4X[25Xgap>[125X [27XH := NilpotentQuotient( G, [x] );[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 12, 0, 5, 10, 2, 0, 30, [128X[104X
    [4X[28X  5, 2, 5, 5, 5, 5 ][128X[104X
    [4X[25Xgap>[125X [27XLeftNormedComm( [ H.1,H.2,H.2,H.2,H.2 ] );[127X[104X
    [4X[28Xid[128X[104X
    [4X[25Xgap>[125X [27Xh := LeftNormedComm( [ H.2,H.1,H.1,H.1,H.1 ] );[127X[104X
    [4X[28Xg9*g10^2*g11^10*g12^5*g13^2*g14^8*g15*g16^6*g17^10*g18*g20^4*g21^4*g22^2*g23^2[128X[104X
    [4X[25Xgap>[125X [27XOrder( h );[127X[104X
    [4X[28X60[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe   previous   calculation   shows  that  in  a  nilpotent  group  without
  [22X2,3,5[122X-torsion a right 4-Engel element is left 4-Engel.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrengel := LeftNormedComm( [a,x,x,x,x,x] );[127X[104X
    [4X[28XComm( a, x, x, x, x, x )[128X[104X
    [4X[25Xgap>[125X [27XG := rec( generators := et, relations := [rengel] );[127X[104X
    [4X[28Xrec( generators := [ a, b, x ], relations := [ Comm( a, x, x, x, x, x ) ] )[128X[104X
    [4X[25Xgap>[125X [27XH := NilpotentQuotient( G, [x], 9 );[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 30, [128X[104X
    [4X[28X  0, 0, 30, 0, 3, 6, 0, 0, 10, 30, 0, 0, 0, 0, 30, 30, 0, 0, 3, 6, 5, 2, 0, [128X[104X
    [4X[28X  2, 408, 2, 0, 0, 0, 10, 10, 30, 10, 0, 0, 0, 3, 3, 3, 2, 204, 6, 6, 0, 10, [128X[104X
    [4X[28X  10, 10, 2, 2, 2, 0, 300, 0, 0, 18 ][128X[104X
    [4X[25Xgap>[125X [27XLeftNormedComm( [ H.1,H.2,H.2,H.2,H.2,H.2 ] );[127X[104X
    [4X[28Xid[128X[104X
    [4X[25Xgap>[125X [27Xh := LeftNormedComm( [ H.2,H.1,H.1,H.1,H.1,H.1 ] );;[127X[104X
    [4X[25Xgap>[125X [27XOrder( h );[127X[104X
    [4X[28Xinfinity[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally,  we  see  that in a torsion-free group a right 5-Engel element need
  not be a left 5-Engel element.[133X
  
