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<title>GAP (HAP) - Chapter 17: G-Outer Groups</title>
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<div class="ChapSects"><a href="chap17.html#X7D02CE0A83211FB7">17. <span class="Heading"> G-Outer Groups</span></a>
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<h3>17. <span class="Heading"> G-Outer Groups</span></h3>
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<td class="tdleft"><code class="code">GOuterGroup(E,N)</code> <br /> <code class="code">GOuterGroup()</code></p>
<p>Inputs a group E and normal subgroup N. It returns N as a G-outer group where G=E/N.</p>
<p>The function can be used without an argument. In this case an empty outer group C is returned. The components must be set using SetActingGroup(C,G), SetActedGroup(C,N) and SetOuterAction(C,alpha).</td>
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<td class="tdleft"><code class="code">GOuterGroupHomomorphismNC(A,B,phi)</code> <br /> <code class="code">GOuterGroupHomomorphismNC()</code></p>
<p>Inputs G-outer groups A and B with common acting group, and a group homomorphism phi:ActedGroup(A) --> ActedGroup(B). It returns the corresponding G-outer homomorphism PHI:A--> B. No check is made to verify that phi is actually a group homomorphism which preserves the G-action.</p>
<p>The function can be used without an argument. In this case an empty outer group homomorphism PHI is returned. The components must then be set.</td>
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<td class="tdleft"><code class="code">GOuterHomomorphismTester(A,B,phi)</code></p>
<p>Inputs G-outer groups A and B with common acting group, and a group homomorphism phi:ActedGroup(A) --> ActedGroup(B). It tests whether phi is a group homomorphism which preserves the G-action.</p>
<p>The function can be used without an argument. In this case an empty outer group homomorphism PHI is returned. The components must then be set.</td>
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<td class="tdleft"><code class="code">Centre(A)</code></p>
<p>Inputs G-outer group A and returns the group theoretic centre of ActedGroup(A) as a G-outer group.</td>
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<td class="tdleft"><code class="code">DirectProductGog(A,B)</code> <br /> <code class="code">DirectProductGog(Lst)</code></p>
<p>Inputs G-outer groups A and B with common acting group, and returns their group-theoretic direct product as a G-outer group. The outer action on the direct product is the diagonal one.</p>
<p>The function also applies to a list Lst of G-outer groups with common acting group.</p>
<p>For a direct product D constructed using this function, the embeddings and projections can be obtained (as G-outer group homomorphisms) using the functions Embedding(D,i) and Projection(D,i).</td>
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