Here we include a list of all functions that are provided to the User.
The following functions are designed to improve the User experience when working or experimenting with wreath products of finite groups and their elements. Most functions are about presenting elements in an intuitive way and giving access to useful information.
The main way for the User to look at a wreath product in a "human-readable" way is by using an isomorphism from a specialised wreath product representation to a generic representation.
‣ IsomorphismWreathProduct( G ) | ( operation ) |
returns an isomorphism from a specialized wreath product G to a generic wreath product.
gap> K := AlternatingGroup(5);; gap> H := SymmetricGroup(4);; gap> G := WreathProduct(K, H); <permutation group of size 311040000 with 10 generators> gap> iso := IsomorphismWreathProduct(G);; gap> W := Image(iso); <group of size 311040000 with 4 generators>
For an overview on wreath product representations in GAP see 5.1-1.
In the background, it uses the low-level functions ListWreathProductElement and WreathProductElementList and wraps the IsList representations into IsWreathProductElement representations.
For performant code, we recommend to use these low-level functions instead of IsomorphismWreathProduct. All functions for IsWreathProductElement also work on IsList objects that represent a wreath product element. However, it is not checked that the IsList object actually represents a wreath product element.
The following functions give access to components of wreath products and their elements.
‣ ComponentsOfWreathProduct( W ) | ( function ) |
returns a list of two groups [K, H], where W = K wr H. The argument W must be a wreath product (see 2.1).
‣ TopGroupOfWreathProduct( W ) | ( function ) |
returns a group, namely the top group \(\langle 1_K \rangle^m \times H\) of the wreath product \(W = K \wr H\) (see 2.1).
‣ BaseGroupOfWreathProduct( W[, i] ) | ( function ) |
returns a group, namely the base group \(K^m \times \langle 1_H\) of the wreath product \(W = K \wr H\). If the optional argument i is provided, the function returns the i-th factor of the base group of W (see 2.1).
‣ TopComponentOfWreathProductElement( x ) | ( function ) |
returns a group element, namely the top component of x. The argument x must be a wreath product element (see 2.1).
‣ BaseComponentOfWreathProductElement( x[, i] ) | ( function ) |
returns a group element, namely the base component of x. If the optional argument i is provided, the function returns the i-th base component of x. The argument x must be a wreath product element and the optional argument i must be an integer (see 2.1).
The following functions give access to important properties of wreath product elements.
‣ Territory( x ) | ( attribute ) |
returns a list, namely the territory of x. The argument x must be a wreath product element (see 2.2).
‣ IsWreathCycle( x ) | ( attribute ) |
returns true or false. Tests whether x is a wreath cycle. The argument x must be a wreath product element (see 2.2).
‣ IsSparseWreathCycle( x ) | ( attribute ) |
returns true or false. Tests whether x is a sparse wreath cycle. The argument x must be a wreath product element (see 2.3).
‣ WreathCycleDecomposition( x ) | ( attribute ) |
returns a list containing wreath cycles, namely the wreath cycle decomposition of x. The argument x must be a wreath product element (see 2.2).
‣ Yade( x[, i] ) | ( attribute ) |
returns a group element, namely the yade of the wreath cycle x evaluated at the smallest territory point. If the optional argument i is provided, the function returns the yade evaluated at the point i. The argument x must be a wreath cycle and the optional argument i must be an integer from the territory of x (see 2.3)
‣ ViewObj( x ) | ( method ) |
‣ PrintObj( x ) | ( method ) |
‣ Display( x[, optrec] ) | ( method ) |
Wreath product elements are viewed, printed and displayed (see Section Reference: View and Print for the distinctions between these operations) as generic wreath product elements (see Section 2.1). For an example of the distinctions and outputs see 3.2.
The method Display allows an optional argument optrec which must be a record and modifies the display output for the execution of a single instance of the command.
For modifying the display output globally for all subsequent executions of Display see SetDisplayOptionsForWreathProductElements (4.4-3).
The following components of optrec are supported. Note, that in the following labels refer to the the printing output \(1, /ldots, m\)
and top
as seen in the tutorials.
horizontaltrue to use the horizontal printer. DEFAULT
false to use the vertical printer.
labelstrue to print labels. DEFAULT
false to suppress labels.
labelStyle"none" for labels in normal intensity. DEFAULT
"bold" for labels in increased intensity.
"faint" for labels in decreased intensity.
labelColor"default" for labels in the default GAP output color. DEFAULT
"red" for labels in red color.
"blue" for labels in blue color.
‣ DisplayOptionsForWreathProductElements( ) | ( function ) |
prints the current global display options for wreath product elements.
‣ SetDisplayOptionsForWreathProductElements( optrec ) | ( function ) |
sets the current global display options for wreath product elements.
The argument optrec must be a record with components that are valid display options. (see 4.4) The components for the current global display options are set to the values specified by the components in optrec.
‣ ResetDisplayOptionsForWreathProductElements( ) | ( function ) |
resets the current global display options for wreath product elements to default.
The following functions construct the cycle index polynomial of wreath products in certain actions.
‣ CycleIndexWreathProductImprimitiveAction( K, H ) | ( function ) |
For two permutation groups K and H this function constructs the cycle index polynomial of the wreath product \(K \wr H\) in imprimitive action.
The implementation is based on [P\t37].
‣ CycleIndexWreathProductProductAction( K, H ) | ( function ) |
For two permutation groups K and H this function constructs the cycle index polynomial of the wreath product \(K \wr H\) in product action.
The implementation is based on [HH68] and [PR73].
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