Primitive Permutation Groups
Primitive Permutation Groups
<#Include Label="[1]{primitiv}">

<#Include Label="[2]{primitiv}"> <#Include Label="PrimitiveGroup"> <#Include Label="NrPrimitiveGroups"> <#Include Label="AllPrimitiveGroups"> <#Include Label="OnePrimitiveGroup"> <#Include Label="PrimitiveGroupsIterator">

In the primitive groups are sorted in cohorts according to their socle. For each degree less than 2500, the variable 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 ), the second entry listing the index numbers of the groups in this degree.

For example in degree 49, we have four cohorts with socles (&ZZ; / 7 &ZZ;)^2, L_2(7)^2, A_7^2 and A_{49} respectively. the group PrimitiveGroup(49,36), which is isomorphic to (A_7 \times A_7):2^2, lies in the third cohort with socle (A_7 \times A_7).

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 ] ] ]
Index numbers of primitive groups
<#Include Label="PrimitiveIdentification"> <#Include Label="SimsNo">

The system `Magma` also provides a list of primitive groups (see ). For historical reasons, its indexing up to degree 999 differs from the one used by &GAP;. The variable can be used to obtain this correspondence. The magma index number of the &GAP; group PrimitiveGroup(deg,nr) is stored in the entry PRIMITIVE_INDICES_MAGMA[deg][nr], for degree at most 999.

Vice versa, the group of degree deg with `Magma` index number nr has the &GAP; index

Position(PRIMITIVE_INDICES_MAGMA[deg],nr)
, in particular it can be obtained by the &GAP; command

PrimitiveGroup(deg,Position(PRIMITIVE_INDICES_MAGMA[deg],nr));