GAP (LINS) - Chapter 3: LINS Search

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3 LINS Search
3.1 LINS Search Options

### 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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