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Add InnerAutomorphismGroup attribute for groups #3914

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Mar 6, 2020
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Add InnerAutomorphismGroup attribute for groups #3914

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1 change: 1 addition & 0 deletions doc/ref/grphomom.xml
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Original file line number Diff line number Diff line change
Expand Up @@ -377,6 +377,7 @@ to form groups of automorphisms.
<#Include Label="AutomorphismDomain">
<#Include Label="IsAutomorphismGroup">
<#Include Label="InnerAutomorphismsAutomorphismGroup">
<#Include Label="InnerAutomorphismGroup">
<#Include Label="InducedAutomorphism">

<!-- %% The code for automorphism groups was designed and implemented by Bettina -->
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21 changes: 21 additions & 0 deletions lib/morpheus.gd
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Expand Up @@ -170,6 +170,27 @@ InstallTrueMethod( IsGroupOfAutomorphisms,IsAutomorphismGroup );
##
DeclareAttribute("InnerAutomorphismsAutomorphismGroup",IsGroup);

#############################################################################
##
#A InnerAutomorphismGroup(<G>)
##
## <#GAPDoc Label="InnerAutomorphismGroup">
## <ManSection>
## <Attr Name="InnerAutomorphismGroup" Arg='G'/>
##
## <Description>
## For a group <A>G</A> this attribute stores the group of inner
## automorphisms (automorphisms induced by conjugation) of the original group.
## <Example><![CDATA[
## gap> InnerAutomorphismGroup(SymmetricGroup(5));
## <group with 2 generators>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("InnerAutomorphismGroup", IsGroup);

#############################################################################
##
#F AssignNiceMonomorphismAutomorphismGroup(<autgrp>,<group>) local
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22 changes: 22 additions & 0 deletions lib/morpheus.gi
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Expand Up @@ -2587,6 +2587,28 @@ InstallMethod( InnerAutomorphismsAutomorphismGroup,
end );


#############################################################################
##
#M InnerAutomorphismGroup( <G> )
##
InstallMethod( InnerAutomorphismGroup,
"for groups",
true,
[ IsGroup and HasGeneratorsOfGroup ], 0,
function( G )
local gens, A;
if HasAutomorphismGroup( G ) or IsTrivial( G ) then
return InnerAutomorphismsAutomorphismGroup( AutomorphismGroup( G ) );
fi;
gens:= GeneratorsOfGroup( G );
# get the non-central generators
gens:= Filtered( gens, i -> not ForAll( gens, j -> i*j = j*i ) );
A := Group( List( gens, i -> InnerAutomorphism( G, i ) ) );
SetIsGroupOfAutomorphismsFiniteGroup( A, true );
return A;
end );


#############################################################################
##
#F IsomorphismGroups(<G>,<H>) . . . . . . . . . . isomorphism from G onto H
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5 changes: 4 additions & 1 deletion tst/testinstall/morpheus.tst
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Expand Up @@ -2,11 +2,14 @@
##
## This file tests the automorphism routines
##
#@local a,autd8,d8,g,inn,iso1,iso2,iso3,iso4,p,r,s4
#@local a,autd8,d8,g,inn,inn2,iso1,iso2,iso3,iso4,p,r,s4
gap> START_TEST("morpheus.tst");
gap> g:=Group((1,2,3,4),(1,3));;
gap> inn2:=InnerAutomorphismGroup(g);;
gap> a:=AutomorphismGroup(g);;
gap> inn:=InnerAutomorphismsAutomorphismGroup(a);;
gap> inn = inn2;
true
gap> iso1:=IsomorphismGroups(a,g);;
gap> iso1=fail;
false
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