Document IsFrattiniFree and add methods for computing it

doc/ref/groups.xml

@@ -332,6 +332,7 @@ as they depend on a parameter.
 
 <#Include Label="IsCyclic">
 <#Include Label="IsElementaryAbelian">
+<#Include Label="IsFrattiniFree">
 <#Include Label="IsNilpotentGroup">
 <#Include Label="NilpotencyClassOfGroup">
 <#Include Label="IsPerfectGroup">

lib/grp.gd

@@ -1834,6 +1834,7 @@ DeclareAttribute( "PrefrattiniSubgroup", IsGroup );
 ##  <Description>
 ##  The Frattini subgroup of a group <A>G</A> is the intersection of all
 ##  maximal subgroups of <A>G</A>.
+##  See also <Ref Prop="IsFrattiniFree"/>.
 ##  <Example><![CDATA[
 ##  gap> FrattiniSubgroup(g);
 ##  Group(())
@@ -1845,6 +1846,60 @@ DeclareAttribute( "PrefrattiniSubgroup", IsGroup );
 DeclareAttribute( "FrattiniSubgroup", IsGroup );
 
 
+#############################################################################
+##
+#P  IsFrattiniFree( <G> )
+##
+##  <#GAPDoc Label="IsFrattiniFree">
+##  <ManSection>
+##  <Prop Name="IsFrattiniFree" Arg='G'/>
+##
+##  <Description>
+##  A group is called <E>Frattini-free</E> if its Frattini subgroup
+##  (see <Ref Attr="FrattiniSubgroup"/>) is trivial.
+##  <P/>
+##  Methods for this property are only installed for finite groups.
+##  Note that a finite nilpotent group is Frattini-free if and only if all
+##  of its Sylow subgroups are elementary abelian, and that a finite group
+##  of squarefree order is always Frattini-free.
+##  <Example><![CDATA[
+##  gap> IsFrattiniFree( SymmetricGroup( 4 ) );
+##  true
+##  gap> IsFrattiniFree( CyclicGroup( 6 ) );
+##  true
+##  gap> IsFrattiniFree( CyclicGroup( 4 ) );
+##  false
+##  gap> IsFrattiniFree( QuaternionGroup( 8 ) );
+##  false
+##  ]]></Example>
+##  </Description>
+##  </ManSection>
+##  <#/GAPDoc>
+##
+DeclareProperty( "IsFrattiniFree", IsGroup );
+
+InstallIsomorphismMaintenance( IsFrattiniFree, IsGroup, IsGroup );
+
+InstallTrueMethod( IsFrattiniFree, IsGroup and IsTrivial );
+
+# In an (infinite dimensional) vector space over a field with p elements the
+# hyperplanes intersect trivially.
+InstallTrueMethod( IsFrattiniFree, IsGroup and IsElementaryAbelian );
+
+# A nontrivial finite group has a maximal subgroup, hence its Frattini
+# subgroup is a proper normal subgroup and thus trivial if the group
+# is simple.
+InstallTrueMethod( IsFrattiniFree, IsGroup and IsFinite and IsSimpleGroup );
+
+# For a finite p-group P we have Phi(P) = P'P^p, hence P is Frattini-free
+# if and only if it is elementary abelian; a finite nilpotent group is the
+# direct product of its Sylow subgroups.
+InstallTrueMethod( IsElementaryAbelian,
+    IsGroup and IsFinite and IsPGroup and IsFrattiniFree );
+InstallTrueMethod( IsCommutative,
+    IsGroup and IsFinite and IsNilpotentGroup and IsFrattiniFree );
+
+
 #############################################################################
 ##
 #A  InvariantForm( <D> )

lib/grp.gi

@@ -1641,6 +1641,58 @@ local m;
     return m;
 end);
 
+InstallMethod( FrattiniSubgroup, "for Frattini-free groups",
+            [ IsGroup and IsFrattiniFree ], SUM_FLAGS,
+            TrivialSubgroup );
+
+
+#############################################################################
+##
+#M  IsFrattiniFree( <G> ) . . . . . . .  is the Frattini subgroup trivial ?
+##
+InstallMethod( IsFrattiniFree, "for groups with known Frattini subgroup",
+            [ IsGroup and HasFrattiniSubgroup ], SUM_FLAGS,
+            G -> IsTrivial( FrattiniSubgroup( G ) ) );
+
+InstallMethod( IsFrattiniFree, "for finite nilpotent groups",
+            [ IsGroup and IsFinite and IsNilpotentGroup ],
+function(G)
+    # A finite nilpotent group is the direct product of its Sylow subgroups,
+    # and for a finite p-group P we have Phi(P) = P'P^p. Hence Phi(G) is
+    # trivial if and only if all Sylow subgroups of G are elementary abelian,
+    # i.e., if and only if G is abelian of squarefree exponent.
+    return IsAbelian(G) and IsDuplicateFree(FactorsInt(Exponent(G)));
+end);
+
+InstallMethod( IsFrattiniFree, "generic method for finite groups",
+            [ IsGroup and IsFinite ],
+function(G)
+local n, F;
+    # A group of squarefree order is Frattini-free.
+    n := Size(G);
+    if IsDuplicateFree(FactorsInt(n)) then
+      return true;
+    fi;
+
+    # If N is normal in G then Phi(N) <= Phi(G). Applied to the nilpotent
+    # normal subgroup F = F(G) this shows that G can only be Frattini-free if
+    # F is abelian of squarefree exponent. Deciding this usually is much
+    # cheaper than computing Phi(G).
+    F := FittingSubgroup(G);
+    if not (IsAbelian(F) and IsDuplicateFree(FactorsInt(Exponent(F)))) then
+      return false;
+    fi;
+
+    # if G = F(G), i.e., if G is nilpotent, this criterion is also sufficient
+    if Size(F) = n then
+      return true;
+    fi;
+
+    return IsTrivial(FrattiniSubgroup(G));
+end);
+
+RedispatchOnCondition( IsFrattiniFree, true, [IsGroup], [IsFinite], 0);
+
 
 #############################################################################
 ##

lib/grppcaut.gd

@@ -11,12 +11,7 @@
 
 DeclareGlobalFunction("SpaceAndOrbitStabilizer");
 
-#############################################################################
-##
-#P IsFrattiniFree
-##
-DeclareProperty( "IsFrattiniFree", IsGroup );
-
+# for IsFrattiniFree see grp.gd
 
 DeclareGlobalFunction("AutomorphismGroupNilpotentGroup");
 DeclareGlobalFunction("AutomorphismGroupSolvableGroup");

lib/methwhy.g

@@ -318,6 +318,14 @@ end);
 ##  +IsFinitelyGeneratedGroup
 ##     IsPolycyclicGroup
 ##
+##  +IsFinite
+##  +IsMagmaWithInverses
+##  +IsAssociative
+##  +IsFrattiniFree
+##     IsCommutative
+##     IsMonomialGroup
+##     IsPolycyclicGroup
+##
 ##  ]]></Example>
 ##  </Description>
 ##  </ManSection>

tst/testinstall/opers/IsFrattiniFree.tst

@@ -0,0 +1,104 @@
+gap> START_TEST("IsFrattiniFree.tst");
+
+#
+gap> IsFrattiniFree(TrivialGroup());
+true
+gap> IsFrattiniFree(Group(()));
+true
+
+# abelian groups: Frattini-free iff the exponent is squarefree
+gap> List([1..12], n -> IsFrattiniFree(CyclicGroup(n)));
+[ true, true, true, false, true, true, true, false, false, true, true, false ]
+gap> IsFrattiniFree(AbelianGroup([2,2,3,5]));
+true
+gap> IsFrattiniFree(AbelianGroup([2,2,3,9]));
+false
+
+# p-groups: Frattini-free iff elementary abelian
+gap> IsFrattiniFree(ElementaryAbelianGroup(27));
+true
+gap> IsFrattiniFree(DihedralGroup(8));
+false
+gap> IsFrattiniFree(QuaternionGroup(8));
+false
+gap> IsFrattiniFree(ExtraspecialGroup(27,3));
+false
+
+# nilpotent groups: Frattini-free iff all Sylow subgroups are elementary abelian
+gap> G := DirectProduct(ElementaryAbelianGroup(4), ElementaryAbelianGroup(9));;
+gap> IsNilpotentGroup(G);
+true
+gap> IsFrattiniFree(G);
+true
+gap> G := DirectProduct(ElementaryAbelianGroup(4), CyclicGroup(9));;
+gap> IsNilpotentGroup(G);
+true
+gap> IsFrattiniFree(G);
+false
+
+# groups of squarefree order are Frattini-free
+gap> IsFrattiniFree(SymmetricGroup(3));
+true
+gap> IsFrattiniFree(AlternatingGroup(4));
+true
+gap> IsFrattiniFree(Group((1,2,3,4,5),(2,3,5,4)));  # Frobenius group of order 20
+true
+
+# simple groups are Frattini-free
+gap> IsFrattiniFree(AlternatingGroup(5));
+true
+gap> IsFrattiniFree(PSL(3,2));
+true
+gap> IsFrattiniFree(SL(2,5));
+false
+
+#
+gap> IsFrattiniFree(SymmetricGroup(4));
+true
+gap> IsFrattiniFree(SymmetricGroup(IsPcGroup, 4));
+true
+gap> IsFrattiniFree(GL(2,3));
+false
+gap> IsFrattiniFree(DirectProduct(AlternatingGroup(5), CyclicGroup(4)));
+false
+
+# the property agrees with the definition
+#@if IsPackageMarkedForLoading( "smallgrp", "" )
+gap> ForAll([1..100], n -> ForAll([1..NrSmallGroups(n)],
+>      i -> IsFrattiniFree(SmallGroup(n,i))
+>           = IsTrivial(FrattiniSubgroup(SmallGroup(n,i)))));
+true
+#@fi
+
+# knowing the property makes computing the Frattini subgroup trivial
+gap> G := SymmetricGroup(5);;
+gap> SetIsFrattiniFree(G, true);
+gap> FrattiniSubgroup(G);
+Group(())
+
+# conversely, a known Frattini subgroup decides the property, also for
+# groups which are not known to be finite
+gap> G := FreeGroup(2);;
+gap> SetFrattiniSubgroup(G, TrivialSubgroup(G));
+gap> IsFrattiniFree(G);
+true
+
+# implied properties
+gap> G := Group((1,2,3,4,5,6,7),(2,3,5)(4,7,6));;
+gap> IsFrattiniFree(G);
+true
+gap> HasIsElementaryAbelian(G);
+false
+gap> G := AbelianGroup(IsPermGroup, [7,7]);;
+gap> IsPGroup(G) and IsFrattiniFree(G);
+true
+gap> HasIsElementaryAbelian(G) and IsElementaryAbelian(G);
+true
+gap> G := AbelianGroup(IsPermGroup, [6,6]);;
+gap> IsNilpotentGroup(G) and IsFrattiniFree(G);
+true
+gap> HasIsCommutative(G) and IsCommutative(G);
+true
+
+#
+gap> STOP_TEST("IsFrattiniFree.tst");