<?xml version="1.0" encoding="ISO-8859-1"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>GAP (nq) - Chapter 4: Examples</title> <meta http-equiv="content-type" content="text/html; charset=iso-8859-1" /> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheet" type="text/css" href="manual.css" /> </head> <body> <div class="pcenter"><table class="chlink"><tr><td class="chlink1">Goto Chapter: </td><td><a href="chap0.html">Top</a></td><td><a href="chap1.html">1</a></td><td><a href="chap2.html">2</a></td><td><a href="chap3.html">3</a></td><td><a href="chap4.html">4</a></td><td><a href="chap5.html">5</a></td><td><a href="chapBib.html">Bib</a></td><td><a href="chapInd.html">Ind</a></td></tr></table><br /></div> <p><a id="s0ss0" name="s0ss0"></a></p> <h3>4. Examples</h3> <p><a id="s1ss0" name="s1ss0"></a></p> <h4>4.1 Right Engel elements</h4> <p>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 <a href="chapBib.html#biBNewmanNickel94">[MW94]</a>. 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.</p> <table class="example"> <tr><td><pre> gap> RequirePackage( "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 </pre></td></tr></table> <p>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.</p> <table class="example"> <tr><td><pre> 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 </pre></td></tr></table> <p>The element h has order 4. In a nilpotent group without 2-torsion a right 3-Engel element is left 3-Engel.</p> <table class="example"> <tr><td><pre> 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 </pre></td></tr></table> <p>The previous calculation shows that in a nilpotent group without 2,3,5-torsion a right 4-Engel element is left 4-Engel.</p> <table class="example"> <tr><td><pre> 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 </pre></td></tr></table> <p>Finally, we see that in a torsion-free group a right 5-Engel element need not be a left 5-Engel element.</p> <div class="pcenter"> <table class="chlink"><tr><td><a href="chap0.html">Top of Book</a></td><td><a href="chap3.html">Previous Chapter</a></td><td><a href="chap5.html">Next Chapter</a></td></tr></table> <br /> <div class="pcenter"><table class="chlink"><tr><td class="chlink1">Goto Chapter: </td><td><a href="chap0.html">Top</a></td><td><a href="chap1.html">1</a></td><td><a href="chap2.html">2</a></td><td><a href="chap3.html">3</a></td><td><a href="chap4.html">4</a></td><td><a href="chap5.html">5</a></td><td><a href="chapBib.html">Bib</a></td><td><a href="chapInd.html">Ind</a></td></tr></table><br /></div> </div> <hr /> <p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p> </body> </html>