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<div class="ChapSects"><a href="chap7_mj.html#X85A3F00985453F95">7 <span class="Heading">Iterators</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7BB5350081B27D17">7.1 <span class="Heading">Some iterators for groups and their isomorphisms</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7F8B54D1806C762D">7.1-1 AllIsomorphismsIterator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X831DA5AE8437578F">7.1-2 AllSubgroupsIterator</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X85413EED812C6497">7.2 <span class="Heading">Operations on iterators</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X87395A9181A35301">7.2-1 CartesianIterator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7C95E27987A812EA">7.2-2 UnorderedPairsIterator</a></span>
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<h3>7 <span class="Heading">Iterators</span></h3>
<p><a id="X7BB5350081B27D17" name="X7BB5350081B27D17"></a></p>
<h4>7.1 <span class="Heading">Some iterators for groups and their isomorphisms</span></h4>
<p>The motivation for adding these operations is partly to give a simple example of an iterator for a list that does not yet exist, and need not be created.</p>
<p><a id="X7F8B54D1806C762D" name="X7F8B54D1806C762D"></a></p>
<h5>7.1-1 AllIsomorphismsIterator</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllIsomorphismsIterator</code>( <var class="Arg">G</var>, <var class="Arg">H</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">‣ AllIsomorphismsNumber</code>( <var class="Arg">G</var>, <var class="Arg">H</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">‣ AllIsomorphisms</code>( <var class="Arg">G</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The main <strong class="pkg">GAP</strong> library contains functions producing complete lists of group homomorphisms such as <code class="code">AllHomomorphisms</code>; <code class="code">AllEndomorphisms</code> and <code class="code">AllAutomorphisms</code>. Here we add the missing <code class="code">AllIsomorphisms(G,H)</code> for a list of isomorphisms from <span class="SimpleMath">\(G\)</span> to <span class="SimpleMath">\(H\)</span>. The method is simple -- find one isomorphism <span class="SimpleMath">\(G \to H\)</span> and compose this with all the automorphisms of <span class="SimpleMath">\(G\)</span>. In all these cases it may not be desirable to construct a list of homomorphisms, but just implement an iterator, and that is what is done here. The operation <code class="code">AllIsomorphismsNumber</code> returns the number of isomorphisms iterated over (this is, of course, just the order of the automorphisms group). The operation <code class="code">AllIsomorphisms</code> produces the list or isomorphisms.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := SmallGroup( 6,1);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">iter := AllIsomorphismsIterator( G, s3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NextIterator( iter );</span>
[ f1, f2 ] -> [ (6,7), (5,6,7) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">n := AllIsomorphismsNumber( G, s3 );</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">AllIsomorphisms( G, s3 );</span>
[ [ f1, f2 ] -> [ (6,7), (5,6,7) ], [ f1, f2 ] -> [ (5,7), (5,6,7) ],
[ f1, f2 ] -> [ (5,6), (5,7,6) ], [ f1, f2 ] -> [ (6,7), (5,7,6) ],
[ f1, f2 ] -> [ (5,7), (5,7,6) ], [ f1, f2 ] -> [ (5,6), (5,6,7) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">iter := AllIsomorphismsIterator( G, s3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for h in iter do Print( ImageElm( h, G.1 ) = (6,7), ", " ); od;</span>
true, false, false, true, false, false,
</pre></div>
<p><a id="X831DA5AE8437578F" name="X831DA5AE8437578F"></a></p>
<h5>7.1-2 AllSubgroupsIterator</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllSubgroupsIterator</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The manual entry for the operation <code class="code">AllSubgroups</code> states that it is only intended to be used on small examples in a classroom situation. Access to all subgroups was required by the <strong class="pkg">XMod</strong> package, so this iterator was introduced here. It used the operations <code class="code">LatticeSubgroups(G)</code> and <code class="code">ConjugacyClassesSubgroups(lat)</code>, and then iterates over the entries in these classes.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">c3c3 := Group( (1,2,3), (4,5,6) );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">iter := AllSubgroupsIterator( c3c3 );</span>
<iterator>
<span class="GAPprompt">gap></span> <span class="GAPinput">while not IsDoneIterator(iter) do Print(NextIterator(iter),"\n"); od;</span>
Group( () )
Group( [ (4,5,6) ] )
Group( [ (1,2,3) ] )
Group( [ (1,2,3)(4,5,6) ] )
Group( [ (1,3,2)(4,5,6) ] )
Group( [ (4,5,6), (1,2,3) ] )
</pre></div>
<p><a id="X85413EED812C6497" name="X85413EED812C6497"></a></p>
<h4>7.2 <span class="Heading">Operations on iterators</span></h4>
<p>This section considers ways of producing an iterator from one or more iterators. It may be that operations equivalent to these are available elsewhere in the library -- if so, the ones here can be removed in due course.</p>
<p><a id="X87395A9181A35301" name="X87395A9181A35301"></a></p>
<h5>7.2-1 CartesianIterator</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CartesianIterator</code>( <var class="Arg">iter1</var>, <var class="Arg">iter2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This iterator returns all pairs <span class="SimpleMath">\([x,y]\)</span> where <span class="SimpleMath">\(x\)</span> is the output of a first iterator and <span class="SimpleMath">\(y\)</span> is the output of a second iterator.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">it1 := Iterator( [ 1, 2, 3 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">it2 := Iterator( [ 4, 5, 6 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">iter := CartesianIterator( it1, it2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">while not IsDoneIterator(iter) do Print(NextIterator(iter),"\n"); od;</span>
[ 1, 4 ]
[ 1, 5 ]
[ 1, 6 ]
[ 2, 4 ]
[ 2, 5 ]
[ 2, 6 ]
[ 3, 4 ]
[ 3, 5 ]
[ 3, 6 ]
</pre></div>
<p><a id="X7C95E27987A812EA" name="X7C95E27987A812EA"></a></p>
<h5>7.2-2 UnorderedPairsIterator</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnorderedPairsIterator</code>( <var class="Arg">iter</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This operation returns pairs <span class="SimpleMath">\([x,y]\)</span> where <span class="SimpleMath">\(x,y\)</span> are output from a given iterator <code class="code">iter</code>. Unlike the output from <code class="code">CartesianIterator(iter,iter)</code>, unordered pairs are returned. In the case <span class="SimpleMath">\(L = [1,2,3,\ldots]\)</span> the pairs are ordered as <span class="SimpleMath">\([1,1],[1,2],[2,2],[1,3],[2,3],[3,3],\ldots\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L := [6,7,8,9];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">iterL := IteratorList( L );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pairsL := UnorderedPairsIterator( iterL );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">while not IsDoneIterator(pairsL) do Print(NextIterator(pairsL),"\n"); od;</span>
[ 6, 6 ]
[ 6, 7 ]
[ 7, 7 ]
[ 6, 8 ]
[ 7, 8 ]
[ 8, 8 ]
[ 6, 9 ]
[ 7, 9 ]
[ 8, 9 ]
[ 9, 9 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">iter4 := IteratorList( [ 4 ] );</span>
<iterator>
<span class="GAPprompt">gap></span> <span class="GAPinput">pairs4 := UnorderedPairsIterator(iter4);</span>
<iterator>
<span class="GAPprompt">gap></span> <span class="GAPinput">NextIterator( pairs4 );</span>
[ 4, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsDoneIterator( pairs4 );</span>
true
</pre></div>
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