<html><head><title>[format] 3 Residual Functions</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="C002S000.htm">Previous</a>] [<a href ="C004S000.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <h1>3 Residual Functions</h1><p> <p> <p> <a name = ""></a> <li><code>ResidualWrtFormation( </code><var>G</var><code>, </code><var>F</var><code> ) O</code> <p> Let <var>G</var> be a finite solvable group and <var>F</var> a formation. Then <code>ResidualWrtFormation</code> returns the <var>F</var>-residual subgroup of <var>G</var>. <p> The following special cases have their own functions. <p> <p> <a name = ""></a> <li><code>NilpotentResidual( </code><var>G</var><code> ) A</code> <p> This is the last term of the descending central series of <var>G</var>. <p> <a name = ""></a> <li><code>PResidual( </code><var>G</var><code>, </code><var>p</var><code> ) O</code> <p> This is the smallest normal subgroup of <var>G</var> whose index is a power of the prime <var>p</var>. <p> <a name = ""></a> <li><code>PiResidual( </code><var>G</var><code>, </code><var>primes</var><code> ) O</code> <p> This is the smallest normal subgroup of <var>G</var> whose index is divisible only by primes in the list <var>primes</var>. <p> <a name = ""></a> <li><code>CoprimeResidual( </code><var>G</var><code>, </code><var>primes</var><code> ) O</code> <p> This is the smallest normal subgroup of <var>G</var> whose index is divisible only by primes <strong>not</strong> in the list <code>primes</code>. <p> <a name = ""></a> <li><code>ElementaryAbelianProductResidual( </code><var>G</var><code> ) A</code> <p> This is the smallest normal subgroup of <var>G</var> whose factor group is a direct product of groups of prime order. <p> <p> [<a href = "chapters.htm">Up</a>] [<a href ="C002S000.htm">Previous</a>] [<a href ="C004S000.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <address>format manual<br>February 2003 </address></body></html>
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