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# 9 Some functions for everyday use

### Sections

1. Groups and actions
2. Iterators
3. Lists and Matrices
4. Cyclotomic numbers
5. Filters and Categories

This chapter contains a number of functions that did not fit in anywhere else. Some of them might be useful for other people, too, so they were included here.

## 9.1 Groups and actions

- `OnSubgroups(` subgroup`,` aut`) F`

  For a group G and an automorphism aut of G, `OnSubgroups(`subgroup`,`aut`)` is the image of subgroup under aut

      gap> G:=Group((1,2,3),(2,3));
      Group([ (1,2,3), (2,3) ])
      gap> alpha:=InnerAutomorphism(G,(1,2,3));
      ^(1,2,3)
      gap> OnSubgroups(Subgroup(G,[(2,3)]),alpha);
      Group([ (1,3) ])

- `RepsCClassesGivenOrder(` group`,` order`) O`

  `RepsCClassesGivenOrder(` group`,` order`)` returns all elements of order order up to conjugacy. Note that the representatives are not always the smallest elements of each conjugacy class.

      gap> RepsCClassesGivenOrder(SymmetricGroup(5),2);
      [ (4,5), (2,3)(4,5) ]

  ## 9.2 Iterators

- `CartesianIterator(` tuplelist`) O`

  Returns an iterator for `Cartesian(`tuplelist`)`

- `ConcatenationOfIterators(` iterlist`) F`

  `ConcatenationOfIterators(`iterlist`)` returns an iterator which runs through all iterators in iterlist. Note that the returned iterator loops over the iterators in iterlist sequentially beginning with the first one.

      gap> it:=Iterator([1,2,3]);;
      gap> it2:=CartesianIterator([[9,10],[11]]);;
      gap> cit:=ConcatenationOfIterators([it,it2]);;
      gap> repeat
      > Print(NextIterator(cit),",\c");
      > until IsDoneIterator(cit);
      1,2,3,[ 9, 11 ],[ 10, 11 ],

  ## 9.3 Lists and Matrices

- `IsListOfIntegers(` list`) P`

  `IsListOfIntegers(` list`)` returns `IsSubset(Integers,` list`)` if list is a dense list and `false` otherwise.

- `List2Tuples(` list`,` int`) O`

  If `Size(` list`)` is divisible by int, `List2Tuples(` list`,`int`)` returns a list list2 of size int such that `Concatenation(` list2`)=` list`` and every element of list2 has the same size.

      gap> List2Tuples([1..6],2);
      [ [ 1, 2, 3 ], [ 4, 5, 6 ] ]

- `MatTimesTransMat(` mat`) O`

  does the same as ``mat`*TransposedMat(` mat`)` but uses slightly less space and time for large matrices.

- `PartitionByFunctionNF(` list`,` f`) O`

  `PartitionByFunctionNF(` list`,` f`)` partitions the list list according to the values of the function f defined on list. If f returns `fail` for some element of list, `PartitionByFunctionNF(` list`,` f`)` enters a break loop. Leaving the break loop with 'return;' is safe because `PartitionByFunctionNF` treats `fail` as all other results of f.

- `PartitionByFunction(` list`,` f`) O`

  `PartitionByFunction(` list`,` f`)` partitions the list list according to the values of the function f defined on list. All elements, for which f returns `fail` are omitted, so `PartitionByFunction` does not necessarily return a partition. If `InfoLevel(InfoRDS)`indexInfoRDS@ttInfoRDS is at least 2, the number of elements for which f returns `fail` is shown (if `fail` is returned at all).

      gap> PartitionByFunctionNF([-1..5],x->x^2);
      [ [ 0 ], [ -1, 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ] ]
      gap> test:=function(x)
      > if x>0 then return Sqrt(x);
      >  else return fail;
      > fi;
      > end;
      function( x ) ... end
      gap> PartitionByFunction([-1..5],test);
      [ [ 1 ], [ 4 ], [ 5 ], [ 2 ], [ 3 ] ]
      gap> SetInfoLevel(InfoRDS,2);
      gap> PartitionByFunction([-1..5],test);
      #I  -2-
      [ [ 1 ], [ 4 ], [ 5 ], [ 2 ], [ 3 ] ]
      gap> PartitionByFunctionNF([-1..5],test);
      Error, function returned <fail> called from
      ...
      brk> return;
      [ [ 1 ], [ 4 ], [ 5 ], [ 2 ], [ 3 ], [ -1, 0 ] ]

  ## 9.4 Cyclotomic numbers

- `IsRootOfUnity(` cyc`) P`

  `IsRootOfUnity` tests if a given cyclotomic is actually a root of unity.

- `CoeffList2CyclotomicList(` list`,` root`) O`

  `CoeffList2CyclogomicList(` list`,` root`)` takes a list of integers list and a root of unity root and returns a list list2, where list2[i]=list[i]* root^(i-1).

- `AbssquareInCyclotomics(` list`,` root`) O`

  For a list of integers and a root of unity, `AbssquareInCyclotomics(` list`,` root`)` returns the modulus of `Sum(CoeffList2CyclotomicList(` list`,` root`))`.

- `CycsGivenCoeffSum(` sum`,` root`) O`

  `CycsGivenCoeffSum(` sum`,` root`)` returns all elements of Z[ root ] such that the coefficient sum is sum and all coefficients are non-negative. The returned list has the following form: The cyclotomic numbers are represented by coefficients. CoeffList2CyclotomicList can be used to get the algebraic number represented by list. The list is partitioned into equivalence classes of elements having the same modulus. For each class the modulus is returned. This means that `CycsGivenCoeffSum` returns a list of pairs where the first entry of each pair is the square of the modulus of an element of the second entry. And the second entry is a list of coefficient lists of cyclotomics in Z[ root ] having the coefficient sum sum.

      gap> CycsGivenCoeffSum(3,E(3));
      [ [ 0, [ [ 1, 1, 1 ] ] ],
        [ 3, [ [ 0, 1, 2 ], [ 0, 2, 1 ], [ 1, 0, 2 ], [ 1, 2, 0 ], [ 2, 0, 1 ],
                [ 2, 1, 0 ] ] ], [ 9, [ [ 0, 0, 3 ], [ 0, 3, 0 ], [ 3, 0, 0 ] ] ] ]
      gap> CycsGivenCoeffSum(2,E(2));
      [ [ 0, [ [ 1, 1 ] ] ], [ 4, [ [ 0, 2 ], [ 2, 0 ] ] ] ]

  ## 9.5 Filters and Categories

  The following was originally posted at the GAP forum by Thomas Breuer BreuersAnswer.

  Each filter in GAP is either a simple filter or a meet of filters. For example, `IsInt` and `IsPosRat` are simple filters, and `IsPosInt` is defined as their meet `IsInt and IsPosRat`.

  Each simple filter is of one of the following kinds.

  1. property: Such a filter is an operation, the filter value can be computed. The (unary) methods of this operation must return `true` or `false`, and the return value is stored in the argument, except if the argument is of a basic data type such as cyclotomic (including rationals and integers), finite field element, permutation, or internally represented list --the latter with a few exceptions. Examples of properties are `IsFinite`, `IsAbelian`, `IsSSortedList`.

  2. attribute tester: Such a filter is associated to an operation that has been created via `DeclareAttribute`, in the sense that the value is `true` if and only if a return value for (a unary method of) this operation is stored in the argument. Currently, attribute values are stored in objects in the filter `IsAttributeStoringRep`. Examples of attribute testers are `HasSize`, `HasCentre`, `HasDerivedSubgroup`.

  2.' property tester: Such a filter is similar to an attribute tester, but the associated operation is a property. So property testers can return `true` also if the argument is not in the filter `IsAttributeStoringRep`. Examples of property testers are `HasIsFinite`, `HasIsAbelian`, `HasIsSSortedList`.

  3. category or representation: These filters are not associated to operations, their values cannot be computed but are set upon creation of an object and should not be changed later, such that for a filter of this kind, one can rely on the fact that if the value is `true` then it was `true` already when the object in question was created.

  The distinction between representation and category is intended to express dependency on or independence of the way how the object is stored internally. For example, `IsPositionalObjectRep`, `IsComponentObjectRep`, and `IsInternalRep` are filters of the representation kind; the idea is that such filters are used in low level methods, and that higher level methods can be implemented without referring to these filters.

  Examples of categories are `IsInt`, `IsRat`, `IsPerm`, `IsFFE`, and filters expressing algebraic structures, such as `IsMagma`, `IsMagmaWithOne`, `IsAdditiveMagma`. When one calls such a filter, one can be sure that no computation is triggered. For example, whenever a quotient of two integers is formed, the result is clearly in the filter `IsRat`, but the system also stores the value of `IsInt`, i.e., GAP does not support ``unevaluated rationals'' for which the `IsInt` value is computed on demand and then stored.

  4. other filters: Some filters do not belong to the above kinds, they are not associated to operations but they are intended to be set (or even reset) by the user or by functions also after the creation of objects. Examples are `IsQuickPositionList`, `CanEasilyTestMembership`, `IsHandledByNiceBasis`.

  Each meet of filters can involve computable simple filters (properties, attribute and property testers) and not computable simple filters (categories, representations, other filters). When one calls a meet of two filters then the two filters from which the meet was formed are evaluated (if necessary). So a meet of filters is computable only if at least one computable simple filter is involved.

- `IsComputableFilter(` filter`) F`

  'IsComputableFilter(filter)' returns true if a the filter filter is computable. Filters for which 'IsComputableFilter' returns false may be used in 'DeclareOperation'.

          gap> IsComputableFilter( IsFinite );
          true
          gap> IsComputableFilter( HasSize );
          true
          gap> IsComputableFilter( HasIsFinite );
          true
          gap> IsComputableFilter( IsPositionalObjectRep );
          false
          gap> IsComputableFilter( IsInt );
          false
          gap> IsComputableFilter( IsQuickPositionList );
          false
          gap> IsComputableFilter( IsInt and IsPosRat );
          false
          gap> IsComputableFilter( IsMagma );
          false

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