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<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap4.html">[Previous Chapter]</a>    <a href="chap6.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap5_mj.html">[MathJax on]</a></p>
<p><a id="X8171DAF2833FF728" name="X8171DAF2833FF728"></a></p>
<div class="ChapSects"><a href="chap5.html#X8171DAF2833FF728">5 <span class="Heading">Groups and homomorphisms</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7E21E6D285E6B12C">5.1 <span class="Heading">Functions for groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X80761843831B468E">5.1-1 Comm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X803A050C7A183CCC">5.1-2 IsCommuting</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X87A8F01286548037">5.1-3 ListOfPowers</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X820B71307E41BEE5">5.1-4 GeneratorsAndInverses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X84CF95227F9D562F">5.1-5 UpperFittingSeries</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7FE4848B7DE6B3FD">5.2 <span class="Heading">Left Cosets for Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X8340B4537F17DCD3">5.2-1 LeftCoset</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X793E48267EF5FD77">5.2-2 <span class="Heading">Inverse</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X80A512877F515DE7">5.3 <span class="Heading">Functions for group homomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X80C9A0B583FEA7B9">5.3-1 EpimorphismByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7C705F2A79F8E43C">5.3-2 Pullback</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X78DD2C617B992BE2">5.3-3 CentralProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X801038CB808FC956">5.3-4 IdempotentEndomorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X81FA9E6C7F3B9238">5.3-5 DirectProductOfFunctions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7CB2D5F27F4182AF">5.3-6 DirectProductOfAutomorphismGroups</a></span>
</div></div>
</div>

<h3>5 <span class="Heading">Groups and homomorphisms</span></h3>

<p><a id="X7E21E6D285E6B12C" name="X7E21E6D285E6B12C"></a></p>

<h4>5.1 <span class="Heading">Functions for groups</span></h4>

<p><a id="X80761843831B468E" name="X80761843831B468E"></a></p>

<h5>5.1-1 Comm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Comm</code>( <var class="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This method has been transferred from package <strong class="pkg">ResClasses</strong>.</p>

<p>It provides a method for <code class="code">Comm</code> when the argument is a list (enclosed in square brackets), and calls the function <code class="code">LeftNormedComm</code>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">Comm( [ (1,2), (2,3) ] );</span>
(1,2,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">Comm( [(1,2),(2,3),(3,4),(4,5),(5,6)] );</span>
(1,5,6)
<span class="GAPprompt">gap></span> <span class="GAPinput">Comm(Comm(Comm(Comm((1,2),(2,3)),(3,4)),(4,5)),(5,6));  ## the same</span>
(1,5,6)

</pre></div>

<p><a id="X803A050C7A183CCC" name="X803A050C7A183CCC"></a></p>

<h5>5.1-2 IsCommuting</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCommuting</code>( <var class="Arg">a</var>, <var class="Arg">b</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This function has been transferred from package <strong class="pkg">ResClasses</strong>.</p>

<p>It tests whether two elements in a group commute.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">D12 := DihedralGroup( 12 );</span>
<pc group of size 12 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( D12, "D12" ); </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a := D12.1;;  b := D12.2;;  </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCommuting( a, b );</span>
false

</pre></div>

<p><a id="X87A8F01286548037" name="X87A8F01286548037"></a></p>

<h5>5.1-3 ListOfPowers</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ListOfPowers</code>( <var class="Arg">g</var>, <var class="Arg">exp</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This function has been transferred from package <strong class="pkg">RCWA</strong>.</p>

<p>The operation <code class="code">ListOfPowers(g,exp)</code> returns the list <span class="SimpleMath">[g,g^2,...,g^exp]</span> of powers of the element <span class="SimpleMath">g</span>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">ListOfPowers( 2, 20 );</span>
[ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384,
 32768, 65536, 131072, 262144, 524288, 1048576 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ListOfPowers( (1,2,3)(4,5), 12 );</span>
[ (1,2,3)(4,5), (1,3,2), (4,5), (1,2,3), (1,3,2)(4,5), (),
 (1,2,3)(4,5), (1,3,2), (4,5), (1,2,3), (1,3,2)(4,5), () ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ListOfPowers( D12.2, 6 );</span>
[ f2, f3, f2*f3, f3^2, f2*f3^2, <identity> of ... ]

</pre></div>

<p><a id="X820B71307E41BEE5" name="X820B71307E41BEE5"></a></p>

<h5>5.1-4 GeneratorsAndInverses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsAndInverses</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This function has been transferred from package <strong class="pkg">RCWA</strong>.</p>

<p>This operation returns a list containing the generators of <span class="SimpleMath">G</span> followed by the inverses of these generators.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsAndInverses( D12 );</span>
[ f1, f2, f3, f1, f2*f3^2, f3^2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsAndInverses( SymmetricGroup(5) );     </span>
[ (1,2,3,4,5), (1,2), (1,5,4,3,2), (1,2) ]

</pre></div>

<p><a id="X84CF95227F9D562F" name="X84CF95227F9D562F"></a></p>

<h5>5.1-5 UpperFittingSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UpperFittingSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LowerFittingSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FittingLength</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>These three functions have been transferred from package <strong class="pkg">ResClasses</strong>.</p>

<p>The upper and lower Fitting series and the Fitting length of a solvable group are described here: <span class="URL"><a href="https://en.wikipedia.org/wiki/Fitting_length">https://en.wikipedia.org/wiki/Fitting_length</a></span>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">UpperFittingSeries( D12 );  LowerFittingSeries( D12 );</span>
[ Group([  ]), Group([ f3, f2*f3 ]), Group([ f1, f3, f2*f3 ]) ]
[ D12, Group([ f3 ]), Group([  ]) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FittingLength( D12 );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">S4 := SymmetricGroup( 4 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">UpperFittingSeries( S4 );</span>
[ Group(()), Group([ (1,2)(3,4), (1,4)(2,3) ]), Group([ (1,2)(3,4), (1,4)
  (2,3), (2,4,3) ]), Group([ (3,4), (2,3,4), (1,2)(3,4) ]) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( last, StructureDescription );</span>
[ "1", "C2 x C2", "A4", "S4" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">LowerFittingSeries( S4 );</span>
[ Sym( [ 1 .. 4 ] ), Alt( [ 1 .. 4 ] ), Group([ (1,4)(2,3), (1,3)
 (2,4) ]), Group(()) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( last, StructureDescription );</span>
[ "S4", "A4", "C2 x C2", "1" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FittingLength( S4);</span>
3

</pre></div>

<p><a id="X7FE4848B7DE6B3FD" name="X7FE4848B7DE6B3FD"></a></p>

<h4>5.2 <span class="Heading">Left Cosets for Groups</span></h4>

<p><a id="X8340B4537F17DCD3" name="X8340B4537F17DCD3"></a></p>

<h5>5.2-1 LeftCoset</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftCoset</code>( <var class="Arg">g</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Since <strong class="pkg">GAP</strong> uses right actions by default, the library contains the operation <code class="code">RightCoset(U,g)</code> for constructing the right coset <span class="SimpleMath">Ug</span> of a subgroup <span class="SimpleMath">U ≤ G</span> and an element <span class="SimpleMath">g ∈ G</span>. It has been noted in the reference manual that, by inverting all the elements in <span class="SimpleMath">Ug</span>, the left coset <span class="SimpleMath">g^-1U</span> is obtained.</p>

<p>Just for the sake of completeness, from August 2022 this package provides the operation <code class="code">LeftCoset(g,U)</code> for constructing the left coset <span class="SimpleMath">gU</span>. Users are strongly recommended to continue to use <code class="code">RightCoset</code> for all serious calculations, since left cosets have a much simpler implementation and do not behave exactly like right cosets.</p>

<p>The methods for left cosets which are provided generally work by converting <span class="SimpleMath">gU</span> to <span class="SimpleMath">Ug^-1</span>; applying the equivalent method for right cosets; and, if necessary, converting back again to left cosets.</p>

<p><span class="SimpleMath">G</span> acts on left cosets by <code class="code">OnLeftInverse</code>: <span class="SimpleMath">(gU)^g_0 = g_0^-1*(gU) = (g_0^-1g)U</span>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">a4 := Group( (1,2,3), (2,3,4) );; SetName( a4, "a4" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">k4 := Group( (1,2)(3,4), (1,3)(2,4) );; SetName( k4, "k4" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rc := RightCosets( a4, k4 );</span>
[ RightCoset(k4,()), RightCoset(k4,(2,3,4)), RightCoset(k4,(2,4,3)) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">lc := LeftCosets( a4, k4 );</span>
[ LeftCoset((),k4), LeftCoset((2,4,3),k4), LeftCoset((2,3,4),k4) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AsSet( lc[2] );</span>
[ (2,4,3), (1,2,3), (1,3,4), (1,4,2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">LeftCoset( (1,4,2), k4 ) = lc[2];</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Representative( lc[2] );</span>
(2,4,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">ActingDomain( lc[2] );</span>
k4
<span class="GAPprompt">gap></span> <span class="GAPinput">(1,4,3) in lc[3];</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">(1,2,3)*lc[2] = lc[3];</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">lc[2]^(1,3,2) = lc[3];</span>
true

</pre></div>

<p><a id="X793E48267EF5FD77" name="X793E48267EF5FD77"></a></p>

<h5>5.2-2 <span class="Heading">Inverse</span></h5>

<p>The inverse of the left coset <span class="SimpleMath">gU</span> is the right coset <span class="SimpleMath">Ug^-1</span>, and conversely. This is an abuse of the attribute <code class="code">Inverse</code>, since the standard requirement, that <span class="SimpleMath">x*x^-1</span> is an identity, does not hold.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">Inverse( rc[3] ) = lc[3];</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Inverse( lc[2] ) = rc[2];</span>
true

</pre></div>

<p><a id="X80A512877F515DE7" name="X80A512877F515DE7"></a></p>

<h4>5.3 <span class="Heading">Functions for group homomorphisms</span></h4>

<p><a id="X80C9A0B583FEA7B9" name="X80C9A0B583FEA7B9"></a></p>

<h5>5.3-1 EpimorphismByGenerators</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EpimorphismByGenerators</code>( <var class="Arg">G</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This function has been transferred from package <strong class="pkg">RCWA</strong>.</p>

<p>It constructs a group homomorphism which maps the generators of <span class="SimpleMath">G</span> to those of <span class="SimpleMath">H</span>. Its intended use is when <span class="SimpleMath">G</span> is a free group, and a warning is printed when this is not the case. Note that anything may happen if the resulting map is not a homomorphism!</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">G := Group( (1,2,3), (3,4,5), (5,6,7), (7,8,9) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">phi := EpimorphismByGenerators( FreeGroup("a","b","c","d"), G );</span>
[ a, b, c, d ] -> [ (1,2,3), (3,4,5), (5,6,7), (7,8,9) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PreImagesRepresentativeNC( phi, (1,2,3,4,5,6,7,8,9) );</span>
d*c*b*a
<span class="GAPprompt">gap></span> <span class="GAPinput">a := G.1;; b := G.2;; c := G.3;; d := G.4;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">d*c*b*a;</span>
(1,2,3,4,5,6,7,8,9)
<span class="GAPprompt">gap></span> <span class="GAPinput">## note that it is easy to produce nonsense: </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">epi := EpimorphismByGenerators( Group((1,2,3)), Group((8,9)) );</span>
Warning: calling GroupHomomorphismByImagesNC without checks
[ (1,2,3) ] -> [ (8,9) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGroupHomomorphism( epi );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Image( epi, (1,2,3) );                                            </span>
()
<span class="GAPprompt">gap></span> <span class="GAPinput">Image( epi, (1,3,2) );</span>
(8,9)

</pre></div>

<p><a id="X7C705F2A79F8E43C" name="X7C705F2A79F8E43C"></a></p>

<h5>5.3-2 Pullback</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pullback</code>( <var class="Arg">hom1</var>, <var class="Arg">hom2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PullbackInfo</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If <span class="SimpleMath">ϕ_1 : G_1 -> H</span> and <span class="SimpleMath">ϕ_2 : G_2 -> H</span> are two group homomorphisms with the same range, then their <em>pullback</em> is the subgroup of <span class="SimpleMath">G_1 × G_2</span> consisting of those elements <span class="SimpleMath">(g_1,g_2)</span> such that <span class="SimpleMath">ϕ_1 g_1 = ϕ_2 g_2</span>.</p>

<p>The attribute <code class="code">PullbackInfo</code> of a pullback group <code class="code">P</code> is similar to <code class="code">DirectProductInfo</code> for a direct product of groups. Its value is a record with the following components:</p>


<dl>
<dt><strong class="Mark"><code class="code">directProduct</code></strong></dt>
<dd><p>the direct product <span class="SimpleMath">G_1 × G_2</span>, and</p>

</dd>
<dt><strong class="Mark"><code class="code">projections</code></strong></dt>
<dd><p>a list with the two projections onto <span class="SimpleMath">G_1</span> and <span class="SimpleMath">G_2</span>.</p>

</dd>
</dl>
<p>There are no embeddings in this record, but it is possible to use the embeddings into the direct product, see <code class="func">Embedding</code> (<a href="/home/runner/gap/doc/ref/chap32.html#X86452F8587CBAEA0"><span class="RefLink">Reference: Embedding</span></a>).</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">s4 := Group( (1,2),(2,3),(3,4) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s3 := Group( (5,6),(6,7) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c3 := Subgroup( s3, [ (5,6,7) ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := GroupHomomorphismByImages( s4, s3, </span>
<span class="GAPprompt">></span> <span class="GAPinput">            [(1,2),(2,3),(3,4)], [(5,6),(6,7),(5,6)] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">i := GroupHomomorphismByImages( c3, s3, [(5,6,7)], [(5,6,7)] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Pfi := Pullback( f, i );</span>
Group([ (2,3,4)(5,7,6), (1,2)(3,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( Pfi );</span>
"A4"
<span class="GAPprompt">gap></span> <span class="GAPinput">info := PullbackInfo( Pfi );</span>
rec( directProduct := Group([ (1,2), (2,3), (3,4), (5,6,7) ]), 
  projections := [ [ (2,3,4)(5,7,6), (1,2)(3,4) ] -> [ (2,3,4), (1,2)(3,4) ], 
      [ (2,3,4)(5,7,6), (1,2)(3,4) ] -> [ (5,7,6), () ] ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">g := (1,2,3)(5,6,7);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( info!.projections[1], g );</span>
(1,2,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( info!.projections[2], g );</span>
(5,6,7) 
<span class="GAPprompt">gap></span> <span class="GAPinput">dp := info!.directProduct;; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a := ImageElm( Embedding( dp, 1 ), (1,4,3) );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">b := ImageElm( Embedding( dp, 2 ), (5,7,6) );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a*b in Pfi;</span>
true

</pre></div>

<p><a id="X78DD2C617B992BE2" name="X78DD2C617B992BE2"></a></p>

<h5>5.3-3 CentralProduct</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CentralProduct</code>( <var class="Arg">G1</var>, <var class="Arg">G2</var>, <var class="Arg">Z1</var>, <var class="Arg">Phi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CentralProductInfo</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This function was added by Thomas Breuer, following discussions with Hongyi Zhao (see <span class="URL"><a href="https://github.com/gap-packages/hap/issues/73">https://github.com/gap-packages/hap/issues/73</a></span>).</p>

<p>Let <var class="Arg">G1</var> and <var class="Arg">G2</var> be two groups, <var class="Arg">Z1</var> be a central subgroup of <var class="Arg">G1</var>, and <var class="Arg">Phi</var> be an isomorphism from <var class="Arg">Z1</var> to a central subgroup of <var class="Arg">G2</var>. The <em>central product</em> defined by these arguments is the factor group of the direct product of <var class="Arg">G1</var> and <var class="Arg">G2</var> by the central subgroup <span class="SimpleMath">{ (z, (<var class="Arg">Phi</var>(z))^-1) : z ∈ <var class="Arg">Z1</var> }</span>.</p>

<p>The attribute <code class="func">CentralProductInfo</code> of a group <span class="SimpleMath">G</span> that has been created by <code class="func">CentralProduct</code> is similar to <code class="func">PullbackInfo</code> (<a href="chap5.html#X7C705F2A79F8E43C"><span class="RefLink">5.3-2</span></a>) for pullback groups. Its value is a record with the following components.</p>


<dl>
<dt><strong class="Mark"><code class="code">projection</code></strong></dt>
<dd><p>the epimorphism from the direct product of <var class="Arg">G1</var> and <var class="Arg">G2</var> to <span class="SimpleMath">G</span>, and</p>

</dd>
<dt><strong class="Mark"><code class="code">phi</code></strong></dt>
<dd><p>the map <var class="Arg">Phi</var>.</p>

</dd>
</dl>
<p>Note that one can access the direct product as the <code class="func">Source</code> (<a href="/home/runner/gap/doc/ref/chap32.html#X7DE8173F80E07AB1"><span class="RefLink">Reference: Source</span></a>) value of the <code class="code">projection</code> map, and one can access <var class="Arg">G1</var> and <var class="Arg">G2</var> as the two embeddings of this direct product, see <code class="func">Embedding</code> (<a href="/home/runner/gap/doc/ref/chap32.html#X86452F8587CBAEA0"><span class="RefLink">Reference: Embedding</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g1 := DihedralGroup( 8 );</span>
<pc group of size 8 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">c1 := Centre( g1 );</span>
Group([ f3 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">cp1 := CentralProduct( g1, g1, c1, IdentityMapping( c1 ) );</span>
Group([ f1, f2, f5, f3, f4, f5 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IdGroup( cp1 ) = IdGroup( ExtraspecialGroup( 2^5, "+" ) );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">g2 := QuaternionGroup( 8 );</span>
<pc group of size 8 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">c2 := Centre( g2 );</span>
Group([ y2 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">cp2 := CentralProduct( g2, g2, c2, IdentityMapping( c2 ) );</span>
Group([ f1, f2, f5, f3, f4, f5 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IdGroup( cp2 ) = IdGroup( ExtraspecialGroup( 2^5, "+" ) );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">info2 := CentralProductInfo( cp2 );</span>
rec( phi := IdentityMapping( Group([ y2 ]) ), 
  projection := [ f1, f2, f3, f4, f5, f6 ] -> [ f1, f2, f5, f3, f4, f5 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">Source( Embedding( Source( info2.projection ), 1 ) ) = g2;</span>
true
</pre></div>

<p><a id="X801038CB808FC956" name="X801038CB808FC956"></a></p>

<h5>5.3-4 IdempotentEndomorphisms</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdempotentEndomorphisms</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdempotentEndomorphismsData</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdempotentEndomorphismsWithImage</code>( <var class="Arg">genG</var>, <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>An endomorphism <span class="SimpleMath">f : G -> G</span> is idempotent if <span class="SimpleMath">f^2=f</span>. It has an image <span class="SimpleMath">R leqslant G</span>; is the identity map when restricted to <span class="SimpleMath">R</span>; and has a kernel <span class="SimpleMath">N</span> which has trivial intersection with <span class="SimpleMath">R</span> and has size <span class="SimpleMath">|G|/|R|</span>.</p>

<p>The operation <code class="code">IdempotentEndomorphismsWithImage(genG,R)</code> returns a list of the images of the generating set <code class="code">genG</code> of a group <span class="SimpleMath">G</span> under the idempotent endomorphisms with image <span class="SimpleMath">R</span>.</p>

<p>The attribute <code class="code">IdempotentEndomorphismsData(G)</code> returns a record <code class="code">data</code> with fields <code class="code">data.gens</code>, a fixed generating set for <span class="SimpleMath">G</span>, and <code class="code">data.images</code> a list of the non-empty outputs of <code class="code">IdempotentEndomorphismsWithImage(genG,R)</code> obtained by iterating over all subgroups of <span class="SimpleMath">G</span>.</p>

<p>The operation <code class="code">IdempotentEndomorphisms(G)</code> returns the list of these mappings obtained using <code class="code">IdempotentEndomorphismsData(G)</code>. The first of these is the zero map, the second is the identity.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">gens := [ (1,2,3,4), (1,2)(3,4) ];; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">d8 := Group( gens );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( d8, "d8" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c2 := Subgroup( d8, [ (2,4) ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IdempotentEndomorphismsWithImage( gens, c2 );</span>
[ [ (), (2,4) ], [ (2,4), () ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IdempotentEndomorphismsData( d8 );</span>
rec( gens := [ (1,2,3,4), (1,2)(3,4) ], 
  images := [ [ [ (), () ] ], [ [ (), (2,4) ], [ (2,4), () ] ], 
      [ [ (), (1,3) ], [ (1,3), () ] ], 
      [ [ (), (1,2)(3,4) ], [ (1,2)(3,4), (1,2)(3,4) ] ], 
      [ [ (), (1,4)(2,3) ], [ (1,4)(2,3), (1,4)(2,3) ] ], 
      [ [ (1,2,3,4), (1,2)(3,4) ] ] ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">List( last.images, L -> Length(L) );</span>
[ 1, 2, 2, 2, 2, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IdempotentEndomorphisms( d8 );               </span>
[ [ (1,2,3,4), (1,2)(3,4) ] -> [ (), () ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (2,4) ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (2,4), () ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (1,3) ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,3), () ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (1,2)(3,4) ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,2)(3,4), (1,2)(3,4) ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (1,4)(2,3) ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,4)(2,3), (1,4)(2,3) ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,2,3,4), (1,2)(3,4) ] ]

</pre></div>

<p>The quaternion group <code class="code">q8</code> is an example of a group with a tail: there is only one subgroup in the lattice which covers the identity subgroup. The only idempotent isomorphisms of such groups are the identity mapping and the zero mapping because the only pairs <span class="SimpleMath">N,R</span> are the whole group and the identity subgroup.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">q8 := QuaternionGroup( 8 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IdempotentEndomorphisms( q8 );</span>
[ [ x, y ] -> [ <identity> of ..., <identity> of ... ], [ x, y ] -> [ x, y ] ]

</pre></div>

<p><a id="X81FA9E6C7F3B9238" name="X81FA9E6C7F3B9238"></a></p>

<h5>5.3-5 DirectProductOfFunctions</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DirectProductOfFunctions</code>( <var class="Arg">G</var>, <var class="Arg">H</var>, <var class="Arg">f1</var>, <var class="Arg">f2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Given group homomorphisms <span class="SimpleMath">f_1 : G_1 -> G_2</span> and <span class="SimpleMath">f_2 : H_1 -> H_2</span>, this operation return the product homomorphism <span class="SimpleMath">f_1 × f_2 : G_1 × G_2 -> H_1 × H_2</span>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">c4 := Group( (1,2,3,4) );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c2 := Group( (5,6) );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f1 := GroupHomomorphismByImages( c4, c2, [(1,2,3,4)], [(5,6)] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c3 := Group( (1,2,3) );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c6 := Group( (1,2,3,4,5,6) );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f2 := GroupHomomorphismByImages( c3, c6, [(1,2,3)], [(1,3,5)(2,4,6)] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c4c3 := DirectProduct( c4, c3 ); </span>
Group([ (1,2,3,4), (5,6,7) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">c2c6 := DirectProduct( c2, c6 ); </span>
Group([ (1,2), (3,4,5,6,7,8) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">f := DirectProductOfFunctions( c4c3, c2c6, f1, f2 ); </span>
[ (1,2,3,4), (5,6,7) ] -> [ (1,2), (3,5,7)(4,6,8) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( f, (1,4,3,2)(5,7,6) ); </span>
(1,2)(3,7,5)(4,8,6)

</pre></div>

<p><a id="X7CB2D5F27F4182AF" name="X7CB2D5F27F4182AF"></a></p>

<h5>5.3-6 DirectProductOfAutomorphismGroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DirectProductOfAutomorphismGroups</code>( <var class="Arg">A1</var>, <var class="Arg">A2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <span class="SimpleMath">A_1,A_2</span> be groups of automorphism of groups <span class="SimpleMath">G_1,G_2</span> respectively. The output of this function is a group <span class="SimpleMath">A_1 × A_2</span> of automorphisms of <span class="SimpleMath">G_1 × G_2</span>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">c9 := Group( (1,2,3,4,5,6,7,8,9) );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ac9 := AutomorphismGroup( c9 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">q8 := QuaternionGroup( IsPermGroup, 8 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">aq8 := AutomorphismGroup( q8 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A := DirectProductOfAutomorphismGroups( ac9, aq8 );</span>
<group with 5 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">genA := GeneratorsOfGroup( A );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := Source( genA[1] );</span>
Group([ (1,2,3,4,5,6,7,8,9), (10,14,12,16)(11,17,13,15), (10,11,12,13)
(14,15,16,17) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">a := genA[1]*genA[5];  </span>
[ (1,2,3,4,5,6,7,8,9), (10,14,12,16)(11,17,13,15), (10,11,12,13)(14,15,16,17) 
 ] -> [ (1,3,5,7,9,2,4,6,8), (10,16,12,14)(11,15,13,17), 
  (10,11,12,13)(14,15,16,17) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( a, (1,9,8,7,6,5,4,3,2)(10,14,12,16)(11,17,13,15) );</span>
(1,8,6,4,2,9,7,5,3)(10,16,12,14)(11,15,13,17)

</pre></div>


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