<p>A subgroup <span class="Math">F\le\mathrm{Aut}(B_{d,k})</span> satifies the compatibility condition (C) if and only if <span class="Math">\mathrm{U}_{k}(F)</span> is locally action isomorphic to <span class="Math">F</span>, see <a href="chapBib.html#biBTor20">[Tor20, Proposition 3.8]</a>. The term <em>compatibility</em> comes from the following translation of this condition into properties of the <span class="Math">(k-1)</span>-local actions of elements of <span class="Math">F</span>: The group <span class="Math">F</span> satisfies (C) if and only if</p>
<p>This section is concerned with testing compatibility of two given elements (see <code class="func">AreCompatibleBallElements</code> (<a href="chap3.html#X84AA17C28591AC1F"><span class="RefLink">3.2-1</span></a>)) and finding an/all elements that is/are compatible with a given one (see <code class="func">CompatibleBallElement</code> (<a href="chap3.html#X8206962582F9C96A"><span class="RefLink">3.2-2</span></a>), <code class="func">CompatibilitySet</code> (<a href="chap3.html#X7ECC9E9982597E12"><span class="RefLink">3.2-3</span></a>)).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AreCompatibleBallElements</code>( <var class="Arg">d</var>, <var class="Arg">k</var>, <var class="Arg">aut1</var>, <var class="Arg">aut2</var>, <var class="Arg">dir</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> if <var class="Arg">aut1</var> and <var class="Arg">aut2</var> are compatible with each other in direction <var class="Arg">dir</var>, and <code class="keyw">false</code> otherwise.</p>
<p>The arguments of this method are a degree <var class="Arg">d</var> <span class="Math">\in\mathbb{N}_{\ge 3}</span>, a radius <var class="Arg">k</var> <span class="Math">\in\mathbb{N}</span>, two automorphisms <var class="Arg">aut1</var>, <var class="Arg">aut2</var> <span class="Math">\in\mathrm{Aut}(B_{d,k})</span>, and a direction <var class="Arg">dir</var> <span class="Math">\in</span><code class="code">[1..d]</code>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CompatibleBallElement</code>( <var class="Arg">F</var>, <var class="Arg">aut</var>, <var class="Arg">dir</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: an element of <var class="Arg">F</var> that is compatible with <var class="Arg">aut</var> in direction <var class="Arg">dir</var> if one exists, and <code class="keyw">fail</code> otherwise.</p>
<p>The arguments of this method are a local action <var class="Arg">F</var> <span class="Math">\le\mathrm{Aut}(B_{d,k})</span>, an element <var class="Arg">aut</var> <span class="Math">\in</span> <var class="Arg">F</var>, and a direction <var class="Arg">dir</var> <span class="Math">\in</span><code class="code">[1..d]</code>.</p>
<dl>
<dt><strong class="Mark">for the arguments <var class="Arg">F</var>, <var class="Arg">aut</var>, <varclass="Arg">dir</var></strong></dt>
<dd><p>Returns: the list of elements of <var class="Arg">F</var> that are compatible with <var class="Arg">aut</var> in direction <var class="Arg">dir</var>.</p>
<p>The arguments of this method are a local action <var class="Arg">F</var> of <span class="Math">\le\mathrm{Aut}(B_{d,k})</span>, an automorphism <var class="Arg">aut</var> <span class="Math">\in F</span>, and a direction <var class="Arg">dir</var> <span class="Math">\in</span><code class="code">[1..d]</code>.</p>
</dd>
<dt><strong class="Mark">for the arguments <var class="Arg">F</var>, <var class="Arg">aut</var>, <varclass="Arg">dirs</var></strong></dt>
<dd><p>Returns: the list of elements of <var class="Arg">F</var> that are compatible with <var class="Arg">aut</var> in all directions of <var class="Arg">dirs</var>.</p>
<p>The arguments of this method are a local action <var class="Arg">F</var> of <span class="Math">\le\mathrm{Aut}(B_{d,k})</span>, an automorphism <var class="Arg">aut</var> <span class="Math">\in F</span>, and a sublist of directions <var class="Arg">dirs</var> <span class="Math">\subseteq</span><code class="code">[1..d]</code>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AssembleAutomorphism</code>( <var class="Arg">d</var>, <var class="Arg">k</var>, <var class="Arg">auts</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the automorphism <span class="Math">(</span><code class="code">aut</code><span class="Math">,(</span><var class="Arg">auts</var><span class="Math">[</span><code class="code">i</code><span class="Math">])_{i=1}^{d})</span> of <span class="Math">B_{d,k+1}</span>, where <code class="code">aut</code> is implicit in <span class="Math">(</span><var class="Arg">auts</var><span class="Math">[</span><code class="code">i</code><span class="Math">])_{i=1}^{d}</span>.</p>
<p>The arguments of this method are a degree <var class="Arg">d</var> <span class="Math">\in\mathbb{N}_{\ge 3}</span>, a radius <var class="Arg">k</var> <span class="Math">\in\mathbb{N}</span>, and a list <var class="Arg">auts</var> of <var class="Arg">d</var> automorphisms <span class="Math">(</span><var class="Arg">auts</var><span class="Math">[</span><code class="code">i</code><span class="Math">])_{i=1}^{d}</span> of <span class="Math">B_{d,k}</span> which comes from an element <span class="Math">(</span><code class="code">aut</code><span class="Math">,(</span><var class="Arg">auts</var><span class="Math">[</span><code class="code">i</code><span class="Math">])_{i=1}^{d})</span> of <span class="Math">\mathrm{Aut}(B_{d,k+1})</span>.</p>
<p>Using the methods of Section <a href="chap3.html#X82C839D7794DDBCD"><span class="RefLink">3.2</span></a>, this section provides methods to test groups for the compatibility condition and search for compatible subgroups inside a given group, e.g. <span class="Math">\mathrm{Aut}(B_{d,k})</span>, or with a certain image under some projection.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximalCompatibleSubgroup</code>( <var class="Arg">F</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The local action <span class="Math">C(</span><var class="Arg">F</var><span class="Math">)\le\mathrm{Aut}(B_{d,k})</span>, which is the maximal compatible subgroup of <var class="Arg">F</var>.</p>
<p>The argument of this attribute is a local action <var class="Arg">F</var> <span class="Math">\le\mathrm{Aut}(B_{d,k})</span> (see <code class="func">IsLocalAction</code> (<a href="chap2.html#X7FCF15167D3A44B7"><span class="RefLink">2.1-1</span></a>)).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SatisfiesC</code>( <var class="Arg">F</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> if <var class="Arg">F</var> satisfies the compatibility condition (C), and <code class="keyw">false</code> otherwise.</p>
<p>The argument of this property is a local action <var class="Arg">F</var> <span class="Math">\le\mathrm{Aut}(B_{d,k})</span> (see <code class="func">IsLocalAction</code> (<a href="chap2.html#X7FCF15167D3A44B7"><span class="RefLink">2.1-1</span></a>)).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CompatibleSubgroups</code>( <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the list of all compatible subgroups of <var class="Arg">F</var>.</p>
<p>The argument of this method is a local action <var class="Arg">F</var> <span class="Math">\le\mathrm{Aut}(B_{d,k})</span>. This method calls <code class="code">AllSubgroups</code> on <span class="Math">F</span> and is therefore slow. Use for instructional purposes on small examples only, and use <code class="func">ConjugacyClassRepsCompatibleSubgroups</code> (<a href="chap3.html#X7CEBD97183C3399E"><span class="RefLink">3.3-4</span></a>) or <code class="func">ConjugacyClassRepsCompatibleGroupsWithProjection</code> (<a href="chap3.html#X84F140AC7AD8D9EE"><span class="RefLink">3.3-5</span></a>) for computations.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugacyClassRepsCompatibleSubgroups</code>( <var class="Arg">F</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list of compatible representatives of conjugacy classes of <var class="Arg">F</var> that contain a compatible subgroup.</p>
<p>The argument of this method is a local action <var class="Arg">F</var> of <span class="Math">\mathrm{Aut}(B_{d,k})</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugacyClassRepsCompatibleGroupsWithProjection</code>( <var class="Arg">l</var>, <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list of compatible representatives of conjugacy classes of <span class="Math">\mathrm{Aut}(B_{d,l})</span> that contain a compatible group which projects to <var class="Arg">F</var> <span class="Math">\le\mathrm{Aut}(B_{d,r})</span>.</p>
<p>The arguments of this method are a radius <var class="Arg">l</var> <span class="Math">\in\mathbb{N}</span>, and a local action <var class="Arg">F</var> <span class="Math">\le\mathrm{Aut}(B_{d,k})</span> for some <span class="Math">k\le l</span>.</p>
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