update to LOOPS 3.4.0
These are simply the changes as distributed.
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Glen Whitney
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58 changed files with 17724 additions and 29097 deletions
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@ -120,10 +120,12 @@
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X82373C5479574F22">6.11-3 QuasigroupsUpToIsomorphism</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X8308F38283C61B20">6.11-4 LoopsUpToIsomorphism</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X87677B0787B4461A">6.11-5 AutomorphismGroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X85B3E22679FD8D81">6.11-6 IsomorphicCopyByPerm</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X8121DE3A78795040">6.11-7 IsomorphicCopyByNormalSubloop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7D09D8957E4A0973">6.11-8 Discriminator</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X812F0DEE7C896E18">6.11-9 AreEqualDiscriminators</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7A42812B7B027DD4">6.11-6 QuasigroupIsomorph</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7BD1AC32851286EA">6.11-7 LoopIsomorph</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X85B3E22679FD8D81">6.11-8 IsomorphicCopyByPerm</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X8121DE3A78795040">6.11-9 IsomorphicCopyByNormalSubloop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7D09D8957E4A0973">6.11-10 Discriminator</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X812F0DEE7C896E18">6.11-11 AreEqualDiscriminators</a></span>
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</div></div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7E996BDD81E594F9">6.12 <span class="Heading">Isotopisms</span></a>
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</span>
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@ -265,28 +267,28 @@
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<p>Note how the Cayley table of a subquasigroup is created only upon explicit demand. Also note that changing the names of elements of a subquasigroup (subloop) automatically changes the names of the elements of the parent subquasigroup (subloop). This is because the elements are shared.</p>
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<div class="example"><pre>
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<span class="GAPprompt">gap></span> <span class="GAPinput">M := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.5 ] );
</span>
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<loop of order 3>
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<span class="GAPprompt">gap></span> <span class="GAPinput">[ Parent( S ) = M, Elements( S ), PosInParent( S ) ];
</span>
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[ true, [ l1, l3, l5], [ 1, 3, 5 ] ]
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<span class="GAPprompt">gap></span> <span class="GAPinput">HasCayleyTable( S );
</span>
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false
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<span class="GAPprompt">gap></span> <span class="GAPinput">SetLoopElmName( S, "s" );; Elements( S ); Elements( M );
</span>
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[ s1, s3, s5 ]
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[ s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11, s12 ]
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<span class="GAPprompt">gap></span> <span class="GAPinput">CayleyTable( S );
</span>
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[ [ 1, 3, 5 ], [ 3, 5, 1 ], [ 5, 1, 3 ] ]
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<span class="GAPprompt">gap></span> <span class="GAPinput">LeftSection( S );
</span>
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[ (), (1,3,5), (1,5,3) ]
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<span class="GAPprompt">gap></span> <span class="GAPinput">[ HasCayleyTable( S ), Parent( S ) = M ];
</span>
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[ true, true ]
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<span class="GAPprompt">gap></span> <span class="GAPinput">L := LoopByCayleyTable( CayleyTable( S ) );; Elements( L );
</span>
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[ l1, l2, l3 ]
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<span class="GAPprompt">gap></span> <span class="GAPinput">[ Parent( L ) = L, IsSubloop( M, S ), IsSubloop( M, L ) ];
</span>
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[ true, true, false ]
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<span class="GAPprompt">gap></span> <span class="GAPinput">LeftSection( L );
</span>
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[ (), (1,2,3), (1,3,2) ]
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<div class="example"><pre>
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<span class="GAPprompt">gap></span> <span class="GAPinput">M := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.5 ] );</span>
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<loop of order 3>
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<span class="GAPprompt">gap></span> <span class="GAPinput">[ Parent( S ) = M, Elements( S ), PosInParent( S ) ];</span>
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[ true, [ l1, l3, l5], [ 1, 3, 5 ] ]
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<span class="GAPprompt">gap></span> <span class="GAPinput">HasCayleyTable( S );</span>
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false
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<span class="GAPprompt">gap></span> <span class="GAPinput">SetLoopElmName( S, "s" );; Elements( S ); Elements( M );</span>
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[ s1, s3, s5 ]
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[ s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11, s12 ]
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<span class="GAPprompt">gap></span> <span class="GAPinput">CayleyTable( S );</span>
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[ [ 1, 3, 5 ], [ 3, 5, 1 ], [ 5, 1, 3 ] ]
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<span class="GAPprompt">gap></span> <span class="GAPinput">LeftSection( S );</span>
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[ (), (1,3,5), (1,5,3) ]
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<span class="GAPprompt">gap></span> <span class="GAPinput">[ HasCayleyTable( S ), Parent( S ) = M ];</span>
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[ true, true ]
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<span class="GAPprompt">gap></span> <span class="GAPinput">L := LoopByCayleyTable( CayleyTable( S ) );; Elements( L );</span>
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[ l1, l2, l3 ]
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<span class="GAPprompt">gap></span> <span class="GAPinput">[ Parent( L ) = L, IsSubloop( M, S ), IsSubloop( M, L ) ];</span>
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[ true, true, false ]
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<span class="GAPprompt">gap></span> <span class="GAPinput">LeftSection( L );</span>
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[ (), (1,2,3), (1,3,2) ]
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</pre></div>
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<p><a id="X78ED50F578A88046" name="X78ED50F578A88046"></a></p>
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@ -349,16 +351,16 @@ false
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<p>Here is an example for multiplication groups and inner mapping groups:</p>
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<div class="example"><pre>
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<span class="GAPprompt">gap></span> <span class="GAPinput">M := MoufangLoop(12,1);
</span>
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<Moufang loop 12/1>
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<span class="GAPprompt">gap></span> <span class="GAPinput">LeftSection(M)[2];
</span>
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(1,2)(3,4)(5,6)(7,8)(9,12)(10,11)
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<span class="GAPprompt">gap></span> <span class="GAPinput">Mlt := MultiplicationGroup(M); Inn := InnerMappingGroup(M);
</span>
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<permutation group of size 2592 with 23 generators>
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Group([ (4,6)(7,11), (7,11)(8,10), (2,6,4)(7,9,11), (3,5)(9,11), (8,12,10) ])
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<span class="GAPprompt">gap></span> <span class="GAPinput">Size(Inn);
</span>
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216
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<div class="example"><pre>
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<span class="GAPprompt">gap></span> <span class="GAPinput">M := MoufangLoop(12,1);</span>
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<Moufang loop 12/1>
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<span class="GAPprompt">gap></span> <span class="GAPinput">LeftSection(M)[2];</span>
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(1,2)(3,4)(5,6)(7,8)(9,12)(10,11)
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<span class="GAPprompt">gap></span> <span class="GAPinput">Mlt := MultiplicationGroup(M); Inn := InnerMappingGroup(M);</span>
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<permutation group of size 2592 with 23 generators>
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Group([ (4,6)(7,11), (7,11)(8,10), (2,6,4)(7,9,11), (3,5)(9,11), (8,12,10) ])
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<span class="GAPprompt">gap></span> <span class="GAPinput">Size(Inn);</span>
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216
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</pre></div>
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<p><a id="X7B45C2AF7C2E28AB" name="X7B45C2AF7C2E28AB"></a></p>
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@ -464,16 +466,16 @@ Group([ (4,6)(7,11), (7,11)(8,10), (2,6,4)(7,9,11), (3,5)(9,11), (8,12,10) ])
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<p>Returns: When <var class="Arg">S</var> is a normal subloop of a loop <var class="Arg">Q</var>, returns the natural projection from <var class="Arg">Q</var> onto <var class="Arg">Q</var><span class="SimpleMath">\(/\)</span><var class="Arg">S</var>.</p>
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<div class="example"><pre>
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<span class="GAPprompt">gap></span> <span class="GAPinput">M := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.3 ] );
</span>
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<loop of order 3>
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<span class="GAPprompt">gap></span> <span class="GAPinput">IsNormal( M, S );
</span>
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true
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<span class="GAPprompt">gap></span> <span class="GAPinput">F := FactorLoop( M, S );
</span>
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<loop of order 4>
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<span class="GAPprompt">gap></span> <span class="GAPinput">NaturalHomomorphismByNormalSubloop( M, S );
</span>
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MappingByFunction( <loop of order 12>, <loop of order 4>,
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function( x ) ... end )
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<div class="example"><pre>
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<span class="GAPprompt">gap></span> <span class="GAPinput">M := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.3 ] );</span>
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<loop of order 3>
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<span class="GAPprompt">gap></span> <span class="GAPinput">IsNormal( M, S );</span>
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true
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<span class="GAPprompt">gap></span> <span class="GAPinput">F := FactorLoop( M, S );</span>
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<loop of order 4>
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<span class="GAPprompt">gap></span> <span class="GAPinput">NaturalHomomorphismByNormalSubloop( M, S );</span>
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MappingByFunction( <loop of order 12>, <loop of order 4>,
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function( x ) ... end )
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</pre></div>
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<p><a id="X821F40748401D698" name="X821F40748401D698"></a></p>
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@ -607,16 +609,30 @@ MappingByFunction( <loop of order 12>, <loop of order 4>,
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<p>While dealing with Cayley tables, it is often useful to rename or reorder the elements of the underlying quasigroup without changing the isomorphism type of the quasigroups. <strong class="pkg">LOOPS</strong> contains several functions for this purpose.</p>
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<p><a id="X7A42812B7B027DD4" name="X7A42812B7B027DD4"></a></p>
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<h5>6.11-6 QuasigroupIsomorph</h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuasigroupIsomorph</code>( <var class="Arg">Q</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
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<p>Returns: When <var class="Arg">Q</var> is a quasigroup and <var class="Arg">f</var> is a permutation of <span class="SimpleMath">\(1,\dots,|\)</span><var class="Arg">Q</var><span class="SimpleMath">\(|\)</span>, returns the quasigroup defined on the same set as <var class="Arg">Q</var> with multiplication <span class="SimpleMath">\(*\)</span> defined by <span class="SimpleMath">\(x*y = \)</span><var class="Arg">f</var><span class="SimpleMath">\((\)</span><var class="Arg">f</var><span class="SimpleMath">\({}^{-1}(x)\)</span><var class="Arg">f</var><span class="SimpleMath">\({}^{-1}(y))\)</span>.</p>
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<p><a id="X7BD1AC32851286EA" name="X7BD1AC32851286EA"></a></p>
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<h5>6.11-7 LoopIsomorph</h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LoopIsomorph</code>( <var class="Arg">Q</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
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<p>Returns: When <var class="Arg">Q</var> is a loop and <var class="Arg">f</var> is a permutation of <span class="SimpleMath">\(1,\dots,|\)</span><var class="Arg">Q</var><span class="SimpleMath">\(|\)</span> fixing <span class="SimpleMath">\(1\)</span>, returns the loop defined on the same set as <var class="Arg">Q</var> with multiplication <span class="SimpleMath">\(*\)</span> defined by <span class="SimpleMath">\(x*y = \)</span><var class="Arg">f</var><span class="SimpleMath">\((\)</span><var class="Arg">f</var><span class="SimpleMath">\({}^{-1}(x)\)</span><var class="Arg">f</var><span class="SimpleMath">\({}^{-1}(y))\)</span>. If <var class="Arg">f</var><span class="SimpleMath">\((1)=c\ne 1\)</span>, the isomorphism <span class="SimpleMath">\((1,c)\)</span> is applied after <var class="Arg">f</var>.</p>
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<p><a id="X85B3E22679FD8D81" name="X85B3E22679FD8D81"></a></p>
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<h5>6.11-6 IsomorphicCopyByPerm</h5>
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<h5>6.11-8 IsomorphicCopyByPerm</h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphicCopyByPerm</code>( <var class="Arg">Q</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
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<p>Returns: When <var class="Arg">Q</var> is a quasigroup and <var class="Arg">f</var> is a permutation of <span class="SimpleMath">\(1,\dots,|\)</span><var class="Arg">Q</var><span class="SimpleMath">\(|\)</span>, returns a quasigroup defined on the same set as <var class="Arg">Q</var> with multiplication <span class="SimpleMath">\(*\)</span> defined by <span class="SimpleMath">\(x*y = \)</span><var class="Arg">f</var><span class="SimpleMath">\((\)</span><var class="Arg">f</var><span class="SimpleMath">\({}^{-1}(x)\)</span><var class="Arg">f</var><span class="SimpleMath">\({}^{-1}(y))\)</span>. When <var class="Arg">Q</var> is a declared loop, a loop is returned. Consequently, when <var class="Arg">Q</var> is a declared loop and <var class="Arg">f</var><span class="SimpleMath">\((1) = k\ne 1\)</span>, then <var class="Arg">f</var> is first replaced with <var class="Arg">f</var><span class="SimpleMath">\(\circ (1,k)\)</span>, to make sure that the resulting Cayley table is normalized.</p>
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<p>Returns: <code class="code">LoopIsomorphism(<var class="Arg">Q</var>,<var class="Arg">f</var>)</code> if <var class="Arg">Q</var> is a loop, and <code class="code">QuasigroupIsomorphism(<var class="Arg">Q</var>,<var class="Arg">f</var>)</code> if <var class="Arg">Q</var> is a quasigroup.</p>
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<p><a id="X8121DE3A78795040" name="X8121DE3A78795040"></a></p>
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<h5>6.11-7 IsomorphicCopyByNormalSubloop</h5>
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<h5>6.11-9 IsomorphicCopyByNormalSubloop</h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphicCopyByNormalSubloop</code>( <var class="Arg">Q</var>, <var class="Arg">S</var> )</td><td class="tdright">( operation )</td></tr></table></div>
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<p>Returns: When <var class="Arg">S</var> is a normal subloop of a loop <var class="Arg">Q</var>, returns an isomorphic copy of <var class="Arg">Q</var> in which the elements are ordered according to the right cosets of <var class="Arg">S</var>. In particular, the Cayley table of <var class="Arg">S</var> will appear in the top left corner of the Cayley table of the resulting loop.<br /></p>
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@ -625,7 +641,7 @@ MappingByFunction( <loop of order 12>, <loop of order 4>,
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<p><a id="X7D09D8957E4A0973" name="X7D09D8957E4A0973"></a></p>
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<h5>6.11-8 Discriminator</h5>
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<h5>6.11-10 Discriminator</h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Discriminator</code>( <var class="Arg">Q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
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<p>Returns: A data structure with isomorphism invariants of a loop <var class="Arg">Q</var>.</p>
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@ -636,7 +652,7 @@ MappingByFunction( <loop of order 12>, <loop of order 4>,
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<p><a id="X812F0DEE7C896E18" name="X812F0DEE7C896E18"></a></p>
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<h5>6.11-9 AreEqualDiscriminators</h5>
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<h5>6.11-11 AreEqualDiscriminators</h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AreEqualDiscriminators</code>( <var class="Arg">D1</var>, <var class="Arg">D2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
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<p>Returns: <code class="code">true</code> if <var class="Arg">D1</var>, <var class="Arg">D2</var> are equal discriminators for the purposes of isomorphism searches.</p>
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