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<html><head><title>[design] 6 Automorphism groups and isomorphism testing for block designs</tit
<body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP005.htm">Previous</a>] [<a href ="CHAP007.htm">Nex <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 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 consist all the permutations of the points <var>{1,...,<code></code><var>D</var><code>.v</code>}</var> which pre 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> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP005.htm">Previous</a>] [<a href ="CHAP007.htm">Nex <P> <address>design manual<br>November 2024 </address></body></html> ¤ Dauer der Verarbeitung: 0.27 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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