Examples
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 . 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.