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4 Examples
4.1 Right Engel elements

### 4 Examples

#### 4.1 Right Engel elements

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 [NN94]. 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 \(5>n\).

    gap> LoadPackage( "nq" );
    true
    gap> ##  SetInfoLevel( InfoNQ, 1 );
    gap> ##
    gap> ##  setup calculation
    gap> ##
    gap> et := ExpressionTrees( "a", "b", "x" );
    [ a, b, x ]
    gap> a := et[1];; b := et[2];; x := et[3];;
    gap>
    gap> ##
    gap> ##  define the group for n = 2,3,4,5
    gap> ##
    gap>
    gap> rengel := LeftNormedComm( [a,x,x] );
    Comm( a, x, x )
    gap> G := rec( generators := et, relations := [rengel] );
    rec( generators := [ a, b, x ], relations := [ Comm( a, x, x ) ] )
    gap> ## The following is equivalent to:
    gap> ##   NilpotentQuotient( : input_string := NqStringExpTrees( G, [x] ) )
    gap> H := NilpotentQuotient( G, [x] );
    Pcp-group with orders [ 0, 0, 0 ]
    gap> LeftNormedComm( [ H.2,H.1,H.1 ] );
    id
    gap> LeftNormedComm( [ H.1,H.2,H.2 ] );
    id

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 \(H\).

    gap> rengel := LeftNormedComm( [a,x,x,x] );
    Comm( a, x, x, x )
    gap> G := rec( generators := et, relations := [rengel] );
    rec( generators := [ a, b, x ], relations := [ Comm( a, x, x, x ) ] )
    gap> H := NilpotentQuotient( G, [x] );
    Pcp-group with orders [ 0, 0, 0, 0, 0, 4, 2, 2 ]
    gap> LeftNormedComm( [ H.1,H.2,H.2,H.2 ] );
    id
    gap> h := LeftNormedComm( [ H.2,H.1,H.1,H.1 ] );
    g6^2*g7*g8
    gap> Order( h );
    4

The element \(h\) has order \(4\). In a nilpotent group without \(2\)-torsion a right 3-Engel element is left 3-Engel.

    gap> rengel := LeftNormedComm( [a,x,x,x,x] );
    Comm( a, x, x, x, x )
    gap> G := rec( generators := et, relations := [rengel] );
    rec( generators := [ a, b, x ], relations := [ Comm( a, x, x, x, x ) ] )
    gap> H := NilpotentQuotient( G, [x] );
    Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 12, 0, 5, 10, 2, 0, 30,
      5, 2, 5, 5, 5, 5 ]
    gap> LeftNormedComm( [ H.1,H.2,H.2,H.2,H.2 ] );
    id
    gap> h := LeftNormedComm( [ H.2,H.1,H.1,H.1,H.1 ] );
    g9*g10^2*g11^10*g12^5*g13^2*g14^8*g15*g16^6*g17^10*g18*g20^4*g21^4*g22^2*g23^2
    gap> Order( h );
    60

The previous calculation shows that in a nilpotent group without \(2,3,5\)-torsion a right 4-Engel element is left 4-Engel.

    gap> rengel := LeftNormedComm( [a,x,x,x,x,x] );
    Comm( a, x, x, x, x, x )
    gap> G := rec( generators := et, relations := [rengel] );
    rec( generators := [ a, b, x ], relations := [ Comm( a, x, x, x, x, x ) ] )
    gap> H := NilpotentQuotient( G, [x], 9 );
    Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 30,
      0, 0, 30, 0, 3, 6, 0, 0, 10, 30, 0, 0, 0, 0, 30, 30, 0, 0, 3, 6, 5, 2, 0,
      2, 408, 2, 0, 0, 0, 10, 10, 30, 10, 0, 0, 0, 3, 3, 3, 2, 204, 6, 6, 0, 10,
      10, 10, 2, 2, 2, 0, 300, 0, 0, 18 ]
    gap> LeftNormedComm( [ H.1,H.2,H.2,H.2,H.2,H.2 ] );
    id
    gap> h := LeftNormedComm( [ H.2,H.1,H.1,H.1,H.1,H.1 ] );;
    gap> Order( h );
    infinity

Finally, we see that in a torsion-free group a right 5-Engel element need not be a left 5-Engel element.

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