<html><head><title>[FORMAT] 5 Covering Subgroups</title></head>
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<h1>5 Covering Subgroups</h1><p>
<p>
Let \X be a collection of groups closed under taking homomorphic images.
An <strong>\X-covering subgroup</strong> of a group <i>G</i> is a subgroup <i>E</i> satisfying
<p>
(C) E is in X , and EV = U whenever E is contained in U is contained in G with U/V in X.
<p>
It follows from the definition that an \X-covering subgroup <i>E</i> of <i>G</i> is
also \X-covering in every subgroup <i>U</i> of <i>G</i> that contains <i>E</i>, and an
easy argument shows that <i>E</i> is an <strong>\X-projector</strong> of every such <i>U</i>,
i.e., <i>E</i> satisfies
<p>
(P) <i>EK</i>/<i>K</i> is an \X-maximal subgroup of <i>U</i>/<i>K</i> whenever <i>K</i> is
normal in <i>U</i>.
<p>
Gaschütz showed that if <b>F</b> is a locally defined formation,
then every finite solvable group has an <b>F</b>-covering subgroup. Indeed,
locally defined formations are the only formations with this property. For
such formations the <b>F</b>-projectors and <b>F</b>-covering subgroups of a
solvable group coincide and form a single conjugacy class of subgroups.
(See <a href="biblio.htm#DH"><cite>DH</cite></a> for details.)
<p>
<a name = ""></a>
<li><code>CoveringSubgroup1( </code><var>G</var><code>, </code><var>F</var><code> ) O</code>
<a name = ""></a>
<li><code>CoveringSubgroup2( </code><var>G</var><code>, </code><var>F</var><code> ) O</code>
<a name = ""></a>
<li><code>CoveringSubgroupWrtFormation( </code><var>G</var><code>, </code><var>F</var><code> ) O</code>
<p>
If <var>F</var> is a locally defined integrated formation in <font face="Gill Sans,Helvetica,Arial">GAP</font> and if <var>G</var> is
a finite solvable group, then the command <code>CoveringSubgroup1( </code><var>G</var><code>, </code><var>F</var><code> )</code>
returns an <var>F</var>-covering subgroup of <var>G</var>.
The function <code>CoveringSubgroup2</code> uses a different algorithm to compute
<b>F</b>-covering subgroups. The user may choose either function. Experiments with large groups suggest that <code>CoveringSubgroup1</code> is somewhat faster.
<code>CoveringSubgroupWrtFormation</code> checks first to see if either of these
two functions has already computed an <var>F</var>-covering subgroup of <var>G</var> and, if
not, it calls <code>FCoveringGroup1</code> to compute one.
<p>
<p>
Nilpotent-covering subgroups are also called <strong>Carter subgroups</strong>.
<p>
<a name = ""></a>
<li><code>CarterSubgroup( </code><var>G</var><code> ) A</code>
<p>
The command <code>CarterSubgroup( </code><var>G</var><code> )</code> is equivalent to
<code>CoveringSubgroupWrtFormation( </code><var>G</var><code>, Formation( "Nilpotent" ) )</code>.
<p>
<p>
All of these functions call upon <b>F</b>-normalizer algorithms as subroutines.
<p>
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