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<Chapter Label="prim">
<Heading>Primitive Permutation Groups</Heading>
<Section Label="Primitive Permutation Groups">
<Heading>Primitive Permutation Groups</Heading>
<#Include Label="[1]{primitiv}">
<P/>
<#Include Label="[2]{primitiv}">
<#Include Label="PrimitiveGroupsAvailable">
<#Include Label="PrimitiveGroup">
<#Include Label="NrPrimitiveGroups">
<#Include Label="AllPrimitiveGroups">
<#Include Label="OnePrimitiveGroup">
<#Include Label="PrimitiveGroupsIterator">
<ManSection>
<Var Name="COHORTS_PRIMITIVE_GROUPS"/>
<Description>
In <Cite Key="DixonMortimer88"/> the primitive groups are sorted in
<Q>cohorts</Q> according to their socle. For each degree less than 2500, the variable
<Ref Var="COHORTS_PRIMITIVE_GROUPS"/> contains a list of the cohorts
for the primitive groups of this degree. Each cohort is represented by a
list of length 2, the first entry specifies the socle type (see
<Ref BookName="ref" Func="SocleTypePrimitiveGroup"/>), the second
entry listing the index numbers of the groups in this degree.
<P/>
For example in degree 49, we have four cohorts with socles <M>(&ZZ; / 7 &ZZ;)^2</M>,
<M>L_2(7)^2</M>, <M>A_7^2</M> and <M>A_{49}</M> respectively. the group
<C>PrimitiveGroup(49,36)</C>,
which is isomorphic to <M>(A_7 \times A_7):2^2</M>, lies
in the third cohort with socle <M>(A_7 \times A_7)</M>.
<P/>
<Example><![CDATA[
gap> COHORTS_PRIMITIVE_GROUPS[49];
[ [ rec( parameter := 7, series := "Z", width := 2 ),
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33 ] ],
[ rec( parameter := [ 2, 7 ], series := "L", width := 2 ), [ 34 ] ],
[ rec( parameter := 7, series := "A", width := 2 ), [ 35, 36, 37, 38 ] ],
[ rec( parameter := 49, series := "A", width := 1 ), [ 39, 40 ] ] ]
]]></Example>
</Description>
</ManSection>
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Index numbers of primitive groups">
<Heading>Index numbers of primitive groups</Heading>
<#Include Label="PrimitiveIdentification">
<#Include Label="SimsNo">
<ManSection>
<Var Name="PRIMITIVE_INDICES_MAGMA"/>
<Description>
The system <Package>Magma</Package> also provides a list of primitive groups
(see <Cite Key="RoneyDougal02"/>). For historical
reasons, its indexing up to degree 999
differs from the one used by &GAP;. The
variable <Ref Var="PRIMITIVE_INDICES_MAGMA"/>
can be used to obtain this correspondence. The magma index number of
the &GAP; group <C>PrimitiveGroup(<A>deg</A>,<A>nr</A>)</C> is stored in the entry
<C>PRIMITIVE_INDICES_MAGMA[<A>deg</A>][<A>nr</A>]</C>, for degree at most 999.
<P/>
Vice versa, the group of degree <A>deg</A> with <Package>Magma</Package>
index number <A>nr</A> has the &GAP; index
<P/>
<C>Position(PRIMITIVE_INDICES_MAGMA[<A>deg</A>],<A>nr</A>)</C>, in particular
it can be obtained by the &GAP; command
<P/>
<C>PrimitiveGroup(<A>deg</A>,Position(PRIMITIVE_INDICES_MAGMA[<A>deg</A>],<A>nr</A>));</C>
</Description>
</ManSection>
</Section>
</Chapter>
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