<html><head><title>[design] 6 Automorphism groups and isomorphism testing for block designs</title></head>
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<h1>6 Automorphism groups and isomorphism testing for block designs</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP006.htm#SECT001">Computing automorphism groups</a>
<li> <A HREF="CHAP006.htm#SECT002">Testing isomorphism</a>
</ol><p>
<p>
The functions in this chapter depend on nauty <a href="biblio.htm#Nau90"><cite>Nau90</cite></a>, <a href="biblio.htm#MP14"><cite>MP14</cite></a>
or bliss <a href="biblio.htm#JK07"><cite>JK07</cite></a> via the GRAPE package, which must be fully
installed in order for these functions to work.
<p>
<p>
<h2><a name="SECT001">6.1 Computing automorphism groups</a></h2>
<p><p>
<a name = "SSEC001.1"></a>
<li><code>AutGroupBlockDesign( </code><var>D</var><code> )</code>
<p>
This function returns the automorphism group of the block design <var>D</var>.
The <strong>automorphism group</strong> <var>Aut(<var>D</var>)</var> of <var>D</var> is the group consisting of
all the permutations of the points <var>{1,...,<code></code><var>D</var><code>.v</code>}</var> which preserve
the block-multiset of <var>D</var>.
<p>
This function is not yet implemented for non-binary block designs.
<p>
This function can also be called via <code>AutomorphismGroup(</code><var>D</var><code>)</code>.
<p>
<pre>
gap> D:=PGPointFlatBlockDesign(2,3,1);; # projective plane of order 3
gap> Size(AutGroupBlockDesign(D));
5616
</pre>
<p>
<p>
<h2><a name="SECT002">6.2 Testing isomorphism</a></h2>
<p><p>
<a name = "SSEC002.1"></a>
<li><code>IsIsomorphicBlockDesign( </code><var>D1</var><code>, </code><var>D2</var><code> )</code>
<p>
This boolean function returns <code>true</code> if and only if block designs
<var>D1</var> and <var>D2</var> are <strong>isomorphic</strong>, that is, there is a bijection from the
point-set of <var>D1</var> to that of <var>D2</var> which maps the block-multiset of <var>D1</var>
to that of <var>D2</var>.
<p>
This function is not yet implemented for non-binary block designs.
<p>
For pairwise isomorphism testing for three or more binary block designs,
see <a href="CHAP006.htm#SSEC002.2">BlockDesignIsomorphismClassRepresentatives</a>.
<p>
<pre>
gap> D1:=BlockDesign(3,[[1],[1,2,3],[2]]);;
gap> D2:=BlockDesign(3,[[1],[1,2,3],[3]]);;
gap> IsIsomorphicBlockDesign(D1,D2);
true
gap> D3:=BlockDesign(4,[[1],[1,2,3],[3]]);;
gap> IsIsomorphicBlockDesign(D2,D3);
false
gap> # block designs with different numbers of points are not isomorphic
</pre>
<p>
<a name = "SSEC002.2"></a>
<li><code>BlockDesignIsomorphismClassRepresentatives( </code><var>L</var><code> )</code>
<p>
Given a list <var>L</var> of binary block designs, this function returns a list
consisting of pairwise non-isomorphic elements of <var>L</var>, representing all
the isomorphism classes of elements of <var>L</var>. The order of the elements
in the returned list may differ from their order in <var>L</var>.
<p>
<pre>
gap> D1:=BlockDesign(3,[[1],[1,2,3],[2]]);;
gap> D2:=BlockDesign(3,[[1],[1,2,3],[3]]);;
gap> D3:=BlockDesign(4,[[1],[1,2,3],[3]]);;
gap> BlockDesignIsomorphismClassRepresentatives([D1,D2,D3]);
[ rec( isBlockDesign := true, v := 4, blocks := [ [ 1 ], [ 1, 2, 3 ], [ 3 ] ],
isBinary := true ),
rec( isBlockDesign := true, v := 3, blocks := [ [ 1 ], [ 1, 2, 3 ], [ 2 ] ],
isBinary := true ) ]
</pre>
<p>
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<address>design manual<br>November 2024
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