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<p id="mathjaxlink" class="pcenter"><a href="chap4_mj.html">[MathJax on]</a></p>
<p><a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a></p>
<div class="ChapSects"><a href="chap4.html#X7A489A5D79DA9E5C">4 <span class="Heading">Examples</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap4.html#X8638E6CE7B5955FB">4.1 <span class="Heading">Right Engel elements</span></a>
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<h3>4 <span class="Heading">Examples</span></h3>

<p><a id="X8638E6CE7B5955FB" name="X8638E6CE7B5955FB"></a></p>

<h4>4.1 <span class="Heading">Right Engel elements</span></h4>

<p>An old problem in the context of Engel elements is the question: Is a right <span class="SimpleMath">n</span>-Engel element left <span class="SimpleMath">n</span>-Engel? It is known that the answer is no. For details about the history of the problem, see <a href="chapBib.html#biBNewmanNickel94">[NN94]</a>. In this paper the authors show that for <span class="SimpleMath">n&gt;4</span> there are nilpotent groups with right <span class="SimpleMath">n</span>-Engel elements no power of which is a left <span class="SimpleMath">n</span>-Engel element. The insight was based on computations with the ANU NQ which we reproduce here. We also show the cases <span class="SimpleMath">5&gt;n</span>.</p>


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

<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 <span class="SimpleMath">H</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">rengel := LeftNormedComm( [a,x,x,x] );</span>
Comm( a, x, x, x )
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">G := rec( generators := et, relations := [rengel] );</span>
rec( generators := [ a, b, x ], relations := [ Comm( a, x, x, x ) ] )
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">H := NilpotentQuotient( G, [x] );</span>
Pcp-group with orders [ 0, 0, 0, 0, 0, 4, 2, 2 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftNormedComm( [ H.1,H.2,H.2,H.2 ] );</span>
id
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">h := LeftNormedComm( [ H.2,H.1,H.1,H.1 ] );</span>
g6^2*g7*g8
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Order( h );</span>
4
</pre></div>

<p>The element <span class="SimpleMath">h</span> has order <span class="SimpleMath">4</span>. In a nilpotent group without <span class="SimpleMath">2</span>-torsion a right 3-Engel element is left 3-Engel.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">rengel := LeftNormedComm( [a,x,x,x,x] );</span>
Comm( a, x, x, x, x )
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">G := rec( generators := et, relations := [rengel] );</span>
rec( generators := [ a, b, x ], relations := [ Comm( a, x, x, x, x ) ] )
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">H := NilpotentQuotient( G, [x] );</span>
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 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftNormedComm( [ H.1,H.2,H.2,H.2,H.2 ] );</span>
id
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">h := LeftNormedComm( [ H.2,H.1,H.1,H.1,H.1 ] );</span>
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
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Order( h );</span>
60
</pre></div>

<p>The previous calculation shows that in a nilpotent group without <span class="SimpleMath">2,3,5</span>-torsion a right 4-Engel element is left 4-Engel.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">rengel := LeftNormedComm( [a,x,x,x,x,x] );</span>
Comm( a, x, x, x, x, x )
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">G := rec( generators := et, relations := [rengel] );</span>
rec( generators := [ a, b, x ], relations := [ Comm( a, x, x, x, x, x ) ] )
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">H := NilpotentQuotient( G, [x], 9 );</span>
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 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftNormedComm( [ H.1,H.2,H.2,H.2,H.2,H.2 ] );</span>
id
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">h := LeftNormedComm( [ H.2,H.1,H.1,H.1,H.1,H.1 ] );;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Order( h );</span>
infinity
</pre></div>

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


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