<!-- $Id: SSP.xml,v 1.15 2007/12/27 19:33:26 alexk Exp $ --> <!-- ********************SSP******************** --> <Chapter Label="SSP"> <Heading>Strong Shoda pairs</Heading> <Section Label="SSPSSP"> <Heading>Computing strong Shoda pairs</Heading> <ManSection> <Attr Name="StrongShodaPairs" Arg="G" Comm="A list of SSPs representatives realizing Wedderburn components of QG" /> <Returns> A list of pairs of subgroups of the input group. </Returns> <Description> The input should be a finite group <A>G</A>. <P/> Computes a list of representatives of the equivalence classes of <E>strong Shoda pairs</E> (<Ref Sect="SSPDef" />) of a finite group <A>G</A>. <P/> <Example> <![CDATA[ gap> StrongShodaPairs( SymmetricGroup(4) ); [ [ Sym( [ 1 .. 4 ] ), Group([ (1,3)(2,4), (1,4)(2,3), (2,4,3), (1,2) ]) ], [ Sym( [ 1 .. 4 ] ), Group([ (1,3)(2,4), (1,4)(2,3), (2,4,3) ]) ], [ Group([ (1,2)(3,4), (1,3,2,4), (3,4) ]), Group([ (1,2)(3,4), (1,3,2,4) ]) ], [ Group([ (1,2)(3,4), (3,4), (1,3,2,4) ]), Group([ (1,2)(3,4), (3,4) ]) ], [ Group([ (1,4)(2,3), (1,3)(2,4), (2,4,3) ]), Group([ (1,4)(2,3), (1,3)(2,4) ]) ] ] gap> StrongShodaPairs( DihedralGroup(64) ); [ [ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f1, f2 ]) ], [ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f1*f2 ]) ], [ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f2 ]) ], [ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f1 ]) ], [ Group([ f1*f2, f4*f5*f6, f5*f6, f6, f3, f3 ]), Group([ f6, f5, f4, f1*f2 ]) ], [ Group([ f6, f5, f2, f3, f4 ]), Group([ f6, f5 ]) ], [ Group([ f6, f2, f3, f4, f5 ]), Group([ f6 ]) ], [ Group([ f2, f3, f4, f5, f6 ]), Group([ ]) ] ] ]]> </Example> </Description> </ManSection> </Section> <Section Label="IsSSP"> <Heading>Properties related with Shoda pairs</Heading> <ManSection> <Oper Name="IsStrongShodaPair" Arg="G K H" Comm="Is (K,H) a strong Shoda pair of G?" /> <Description> The first argument should be a finite group <A>G</A>, the second one a sugroup <A>K</A> of <A>G</A> and the third one a subgroup of <A>K</A>. <P/> Returns <K>true</K> if (<A>K</A>,<A>H</A>) is a <E>strong Shoda pair</E> (<Ref Sect="SSPDef" />) of <A>G</A>, and <K>false</K> otherwise. <Example> <![CDATA[ gap> G:=SymmetricGroup(3);; K:=Group([(1,2,3)]);; H:=Group( () );; gap> IsStrongShodaPair( G, K, H ); true gap> IsStrongShodaPair( G, G, H ); false gap> IsStrongShodaPair( G, K, K ); false gap> IsStrongShodaPair( G, G, K ); true ]]> </Example> </Description> </ManSection> <ManSection> <Oper Name="IsShodaPair" Arg="G K H" Comm="Is (K,H) a Shoda pair of G?" /> <Description> The first argument should be a finite group <A>G</A>, the second a subgroup <A>K</A> of <A>G</A> and the third one a subgroup of <A>K</A>. <P/> Returns <K>true</K> if (<A>K</A>,<A>H</A>) is a <E>Shoda pair</E> (<Ref Sect="SPDef" />) of <A>G</A>.<P/> Note that every strong Shoda pair is a Shoda pair, but the converse is not true. <Example> <![CDATA[ gap> G:=AlternatingGroup(5);; gap> K:=AlternatingGroup(4);; gap> H := Group( (1,2)(3,4), (1,3)(2,4) );; gap> IsStrongShodaPair( G, K, H ); false gap> IsShodaPair( G, K, H ); true ]]> </Example> </Description> </ManSection> <Alt Only="LaTeX">\newpage</Alt> <ManSection> <Oper Name="IsStronglyMonomial" Arg="G" Comm="Is every irreducible character strongly monomial" /> <Description> The input <A>G</A> should be a finite group. <P/> Returns <K>true</K> if <A>G</A> is a <E>strongly monomial</E> (<Ref Sect="StMon" />) finite group. <Example> <![CDATA[ gap> S4:=SymmetricGroup(4);; gap> IsStronglyMonomial(S4); true gap> G:=SmallGroup(24,3);; gap> IsStronglyMonomial(G); false gap> IsMonomial(G); false gap> G:=SmallGroup(1000,86);; gap> IsMonomial(G); true gap> IsStronglyMonomial(G); false ]]> </Example> </Description> </ManSection> </Section> </Chapter>