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InclusionMappingGroups</code >( <var class="Arg" >G</var >, <var class="Arg" >H</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; MappingToOne</code >( <var class="Arg" >G</var >, <var class="Arg" >H</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >map <code class="code" >incd8</code > is the inclusion of <code class="code" >d8</code > in <code class="code" >d16</code > used in Section <a href="chap3.html#X7B09A28579707CAF" ><span class="RefLink" >3 .4 </span ></a>. <code class="code" >MappingToOne(G,H)</code > maps the whole of <span class="SimpleMath" >G</span > to the identity element in <span class="SimpleMath" >H</span >.</p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >Print( incd8, "\n" );</span >11 ,13 ,15 ,17 )(12 ,14 ,16 ,18 ), (11 ,18 )(12 ,17 )(13 ,16 )(14 ,15 ) ] ->11 ,13 ,15 ,17 )(12 ,14 ,16 ,18 ), (11 ,18 )(12 ,17 )(13 ,16 )(14 ,15 ) ]span class="GAPprompt" >gap></span > <span class="GAPinput" >imd8 := Image( incd8 );;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >MappingToOne( c4, imd8 );</span >11 ,13 ,15 ,17 )(12 ,14 ,16 ,18 ) ] -> [ () ]pre ></div >"X81D29E737F3D4878" name="X81D29E737F3D4878" ></a></p>14 .1 -2 InnerAutomorphismsByNormalSubgroup</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; InnerAutomorphismsByNormalSubgroup</code >( <var class="Arg" >G</var >, <var class="Arg" >N</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >code class="code" >G</code > by the elements of a normal subgroup <code class="code" >N</code > are calculated, often with <code class="code" >G</code > = <code class="code" >N</code >.</p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >autd8 := AutomorphismGroup( d8 );;</span span class="GAPprompt" >gap></span > <span class="GAPinput" >innd8 := InnerAutomorphismsByNormalSubgroup( d8, d8 );;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >GeneratorsOfGroup( innd8 );</span >1 ,2 ,3 ,4 ), ^(1 ,3 ) ]pre ></div >"X7FC631B786C1DC8B" name="X7FC631B786C1DC8B" ></a></p>14 .1 -3 IsGroupOfAutomorphisms</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsGroupOfAutomorphisms</code >( <var class="Arg" >A</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >IsGroupOfAutomorphisms( innd8 );</span pre ></div >"X852BD9CA84C2AFF0" name="X852BD9CA84C2AFF0" ></a></p>14 .2 <span class="Heading" >Abelian Modules</span ></h4>"X806DEFCC859BB4F1" name="X806DEFCC859BB4F1" ></a></p>14 .2 -1 AbelianModuleObject</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; AbelianModuleObject</code >( <var class="Arg" >grp</var >, <var class="Arg" >act</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsAbelianModule</code >( <var class="Arg" >obj</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; AbelianModuleGroup</code >( <var class="Arg" >obj</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; AbelianModuleAction</code >( <var class="Arg" >obj</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >d module constructor <code class="func" >XModByAbelianModule</code > (<a href="chap2.html#X824631577864961E" ><span class="RefLink" >2 .1 -7 </span ></a>).</p>code class="code" >Xabmod</code > is isomorphic to the output from <code class="code" >XModByAutomorphismGroup( k4 );</code >.</p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >x := (6 ,7 )(8 ,9 );; y := (6 ,8 )(7 ,9 );; z := (6 ,9 )(7 ,8 );;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >k4a := Group( x, y );; SetName( k4a, "k4a" );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >gens3a := [ (1 ,2 ), (2 ,3 ) ];;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >s3a := Group( gens3a );; SetName( s3a, "s3a" );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >alpha := GroupHomomorphismByImages( k4a, k4a, [x,y], [y,x] );;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >beta := GroupHomomorphismByImages( k4a, k4a, [x,y], [x,z] );;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >auta := Group( alpha, beta );;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >acta := GroupHomomorphismByImages( s3a, auta, gens3a, [alpha,beta] );;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >abmod := AbelianModuleObject( k4a, acta );;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Xabmod := XModByAbelianModule( abmod );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Display( Xabmod );</span >Source group k4a has generators:6 ,7 )(8 ,9 ), (6 ,8 )(7 ,9 ) ]1 ,2 ), (2 ,3 ) ]source generators to:1 ,2 ) --> { source gens --> [ (6 ,8 )(7 ,9 ), (6 ,7 )(8 ,9 ) ] }2 ,3 ) --> { source gens --> [ (6 ,7 )(8 ,9 ), (6 ,9 )(7 ,8 ) ] }2 automorphisms generate the group of automorphisms.pre ></div >div class="chlinkprevnextbot" > <a href="chap0.html" >[Top of Book]</a> <a href="chap0.html#contents" >[Contents]</a> <a href="chap13.html" >[Previous Chapter]</a> <a href="chap15.html" >[Next Chapter]</a> </div >div class="chlinkbot" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0.html" >Top</a> <a href="chap1.html" >1 </a> <a href="chap2.html" >2 </a> <a href="chap3.html" >3 </a> <a href="chap4.html" >4 </a> <a href="chap5.html" >5 </a> <a href="chap6.html" >6 </a> <a href="chap7.html" >7 </a> <a href="chap8.html" >8 </a> <a href="chap9.html" >9 </a> <a href="chap10.html" >10 </a> <a href="chap11.html" >11 </a> <a href="chap12.html" >12 </a> <a href="chap13.html" >13 </a> <a href="chap14.html" >14 </a> <a href="chap15.html" >15 </a> <a href="chapBib.html" >Bib</a> <a href="chapInd.html" >Ind</a> </div >hr />"foot" >generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc " >GAPDoc2HTML</a></p>body >html > Messung V0.5 in Prozent C=100 H=100 G=100
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