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<?xml version="1.0" encoding="UTF-8" ?>DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd " >html xmlns="http://www.w3.org/1999/xhtml " xml:lang="en" >head >script type="text/javascript" "https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML " >script >title >GAP (corefreesub) - Chapter 2: Obtaining Core-Free Subgroups</title >meta http-equiv="content-type" content="text/html; charset=UTF-8" />meta name="generator" content="GAPDoc2HTML" />link rel="stylesheet" type="text/css" href="manual.css" />script src="manual.js" type="text/javascript" ></script >script type="text/javascript" >overwriteStyle();</script >head >body class="chap2" onload="jscontent()" >div class="chlinktop" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0_mj.html" >Top</a> <a href="chap1_mj.html" >1</a> <a href="chap2_mj.html" >2</a> <a href="chap3_mj.html" >3</a> <a href="chap4_mj.html" >4</a> <a href="chapBib_mj.html" >Bib</a> <a href="chapInd_mj.html" >Ind</a> </div >div class="chlinkprevnexttop" > <a href="chap0_mj.html" >[Top of Book]</a> <a href="chap0_mj.html#contents" >[Contents]</a> <a href="chap1_mj.html" >[Previous Chapter]</a> <a href="chap3_mj.html" >[Next Chapter]</a> </div >"mathjaxlink" class="pcenter" ><a href="chap2.html" >[MathJax off]</a></p>"X78D8803A84DB6CE8" name="X78D8803A84DB6CE8" ></a></p>div class="ChapSects" ><a href="chap2_mj.html#X78D8803A84DB6CE8" >2 <span class="Heading" >Obtaining Core-Free Subgroups</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X79EF503F837B7702" >2.1 <span class="Heading" >Core-Free Subgroups</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7A9042C27F0BA85B" >2.1-1 IsCoreFree</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7D0391668782765C" >2.1-2 CoreFreeConjugacyClassesSubgroups</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X821F9BD687383C1F" >2.1-3 AllCoreFreeSubgroups</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X847B222E84342E88" >2.2 <span class="Heading" >Degrees of Core-Free subgroups</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X844F04457C6EDC67" >2.2-1 CoreFreeDegrees</a></span >div ></div >div >span class="Heading" >Obtaining Core-Free Subgroups</span ></h3>"X79EF503F837B7702" name="X79EF503F837B7702" ></a></p>span class="Heading" >Core-Free Subgroups</span ></h4>"X7A9042C27F0BA85B" name="X7A9042C27F0BA85B" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsCoreFree</code >( <var class="Arg" >G</var >, <var class="Arg" >H</var > )</td ><td class="tdright" >( function )</td ></tr ></table ></div >var class="Arg" >G</var > and one of its subgroups <var class="Arg" >H</var >, it returns whether <var class="Arg" >H</var > is core-free in <var class="Arg" >G</var >.</p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >G := SymmetricGroup(4);; H := Subgroup(G, [(1,3)(2,4)]);;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Core(G,H);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >IsCoreFree(G,H);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >H := Subgroup(G, [(1,4)(2,3), (1,3)(2,4)]);;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >IsCoreFree(G,H);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Core(G,H);# H is a normal subgroup of G, hence it does not have a trivial core</span >pre ></div >"X7D0391668782765C" name="X7D0391668782765C" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ CoreFreeConjugacyClassesSubgroups</code >( <var class="Arg" >G</var > )</td ><td class="tdright" >( function )</td ></tr ></table ></div >var class="Arg" >G</var ></p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >G := SymmetricGroup(4);; dh := DihedralGroup(10);;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >cfccs := CoreFreeConjugacyClassesSubgroups(G);; Size(cfccs);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >cfccs_dh := CoreFreeConjugacyClassesSubgroups(dh);; Size(cfccs_dh);</span >pre ></div >"X821F9BD687383C1F" name="X821F9BD687383C1F" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ AllCoreFreeSubgroups</code >( <var class="Arg" >G</var > )</td ><td class="tdright" >( function )</td ></tr ></table div >var class="Arg" >G</var ></p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >G := SymmetricGroup(4);; dh := DihedralGroup(10);;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >acfs := AllCoreFreeSubgroups(G);; Size(acfs);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >acfs_dh := AllCoreFreeSubgroups(dh);; Size(acfs_dh);</span >pre ></div >"X847B222E84342E88" name="X847B222E84342E88" ></a></p>span class="Heading" >Degrees of Core-Free subgroups</span ></h4>"X844F04457C6EDC67" name="X844F04457C6EDC67" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ CoreFreeDegrees</code >( <var class="Arg" >G</var > )</td ><td class="tdright" >( function )</td ></tr ></table ></div >f <var class="Arg" >G</var >. The degrees of a faithful transitive permutation representation of <var class="Arg" >G</var > are the index of its core-free subgroups.</p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >G := SymmetricGroup(4);; dh := DihedralGroup(10);;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >CoreFreeDegrees(G);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >CoreFreeDegrees(dh);</span >pre ></div >div class="chlinkprevnextbot" > <a href="chap0_mj.html" >[Top of Book]</a> <a href="chap0_mj.html#contents" >[Contents]</a> <a href="chap1_mj.html" >[Previous Chapter]</a> <a href="chap3_mj.html" >[Next Chapter]</a> </div >div class="chlinkbot" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0_mj.html" >Top</a> <a href="chap1_mj.html" >1</a> <a href="chap2_mj.html" >2</a> <a href="chap3_mj.html" >3</a> <a href="chap4_mj.html" >4</a> <a href="chapBib_mj.html" >Bib</a> <a href="chapInd_mj.html" >Ind</a> </div >hr />"foot" >generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc " >GAPDoc2HTML</a></p>body >html >quality 99%
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