<html><head><title>[format] 5 Covering Subgroups</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="C004S000.htm">Previous</a>] [<a href ="C006S000.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <h1>5 Covering Subgroups</h1><p> <p> Let <var><font face="helvetica,arial">X</font></var> be a collection of groups closed under taking homomorphic images. An <strong><var><font face="helvetica,arial">X</font></var>-covering subgroup</strong> of a group <var>G</var> is a subgroup <var>E</var> satisfying <p> (C) <var>E in<font face="helvetica,arial">X</font></var>, and <var>EV = U</var> whenever <var>E leU leG</var> with <var>U/V in <font face="helvetica,arial">X</font></var>. <p> It follows from the definition that an <var><font face="helvetica,arial">X</font></var>-covering subgroup <var>E</var> of <var>G</var> is also <var><font face="helvetica,arial">X</font></var>-covering in every subgroup <var>U</var> of <var>G</var> that contains <var>E</var>, and an easy argument shows that <var>E</var> is an <strong><var><font face="helvetica,arial">X</font></var>-projector</strong> of every such <var>U</var>, i.e., <var>E</var> satisfies <p> (P) <var>EK/K</var> is an <var><font face="helvetica,arial">X</font></var>-maximal subgroup of <var>U/K</var> whenever <var>K</var> is normal in <var>U</var>. <p> Gaschütz showed that if <var><font face="helvetica,arial">F</font></var> is a locally defined formation, then every finite solvable group has an <var><font face="helvetica,arial">F</font></var>-covering subgroup. Indeed, locally defined formations are the only formations with this property. For such formations the <var><font face="helvetica,arial">F</font></var>-projectors and <var><font face="helvetica,arial">F</font></var>-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 <var><font face="helvetica,arial">F</font></var>-covering subgroups. The user may choose either function. Experiments with large groups suggest that CoveringSubgroup1 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, then 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 <var><font face="helvetica,arial">F</font></var>-normalizer algorithms as subroutines. <p> [<a href = "chapters.htm">Up</a>] [<a href ="C004S000.htm">Previous</a>] [<a href ="C006S000.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <address>format manual<br>February 2003 </address></body></html>
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