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3 LINS Search

This chapter is for advanced users and those interested in a brief introduction to the mathematical background of LINS.

(NOTE: The internal options below might change during further development!)

3.1 LINS Search Options

The method LowIndexNormalSubgroupsSearch allows an optional argument opts which must be a record and modifies the search for the execution of only this single command.

The following components of opts are supported.

DoSetParent

true to set parent of every subgroup to the group G that is contained in the root of the search graph. It also sets the property IsNormalInParent to true for every subgroup.

This should be only disabled for debugging or testing purposes.

InitGraph

a function that takes as an argument a LinsGraph gr. It can popullate the record object gr via calls like gr!.NAME = VALUE in order to initialize certain components that might be needed by other functions later on (see option DoTerminate).

DoCut

a function that takes as arguments a LinsGraph gr and a LinsNode rH.

Returns true if subgroups under rH should not be computed, i.e. the branch will be cut under the node rH.

DoTerminate

a function that takes as arguments a LinsGraph gr and two LinsNodes rH and rK. We are currently computing the subgroups under rH and found the normal subgroup rK.

This function may write data to gr!.ComputedNormalSubgroups`. Make sure to initalize this via the option InitGraph (for example to an empty list), since it is not bound by default.

Returns true if the search can be terminated.

UseLIS

false to use the new procedure with GQuotient calls in LINS_FindTQuotients.

true to use the old procedure with a LowIndexSubgroupsFpGroup call in LINS_FindTQuotients.

FilterTQuotients

a function that takes as an argument a LinsGraph gr and a list of targets following the specifications of LINS_FindTQuotients.

The value of targets in a call via LowIndexNormalSubgroupsSearch depends on the value of the option UseLIS.

If UseLIS is false, we have targets = LINS_TargetsQuotient.

If UseLIS is true, we have targets = LINS_TargetsQuotientUseLIS.

Returns a sublist from targets, that will be used by LINS_FindTQuotients.

DoIntersection

a function that takes as an argument a LinsGraph gr, two LinsNodes rH and rK, and a positive integer i. Let \(G\) be the group that is contained in the root of the search graph.

Returns true if the intersection \(U\) of the groups in rH and rK with index \([G : U] = i\) should be computed.

DoPQuotient

a function that takes as an argument a LinsGraph gr, a LinsNode rH and a prime p.

Returns true if p-quotients under rH should be computed for the prime p.

DoPModule

a function that takes as an argument a LinsGraph gr, a LinsNode rH, a prime p and a positive integer i. Let \(G\) be the group that is contained in the root of the search graph.

Returns true if the normal subgroup \(K\) of index \([G : K] = i\) with elementary abelian p-quotient in rH should be computed.

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