  
  [1X4 [33X[0;0YFunctions[133X[101X
  
  [33X[0;0YHere we include a list of all functions that are provided to the User.[133X
  
  [33X[0;0YThe  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.[133X
  
  
  [1X4.1 [33X[0;0YGeneric Wreath Product Representation[133X[101X
  
  [33X[0;0YThe  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.[133X
  
  [1X4.1-1 IsomorphismWreathProduct[101X
  
  [33X[1;0Y[29X[2XIsomorphismWreathProduct[102X( [3XG[103X ) [32X operation[133X
  
  [33X[0;0Yreturns  an  isomorphism  from  a  specialized wreath product [3XG[103X to a generic
  wreath product.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XK := AlternatingGroup(5);;[127X[104X
    [4X[25Xgap>[125X [27XH := SymmetricGroup(4);;[127X[104X
    [4X[25Xgap>[125X [27XG := WreathProduct(K, H);[127X[104X
    [4X[28X<permutation group of size 311040000 with 10 generators>[128X[104X
    [4X[25Xgap>[125X [27Xiso := IsomorphismWreathProduct(G);;[127X[104X
    [4X[25Xgap>[125X [27XW := Image(iso);[127X[104X
    [4X[28X<group of size 311040000 with 4 generators>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor an overview on wreath product representations in [5XGAP[105X see [14X5.1-1[114X.[133X
  
  [33X[0;0YIn  the background, it uses the low-level functions [10XListWreathProductElement[110X
  and  [10XWreathProductElementList[110X  and  wraps  the  [10XIsList[110X  representations into
  [10XIsWreathProductElement[110X representations.[133X
  
  [33X[0;0YFor  performant  code, we recommend to use these low-level functions instead
  of  [10XIsomorphismWreathProduct[110X.  All functions for [10XIsWreathProductElement[110X also
  work  on [10XIsList[110X objects that represent a wreath product element. However, it
  is  not  checked that the [10XIsList[110X object actually represents a wreath product
  element.[133X
  
  
  [1X4.2 [33X[0;0YAccessing Components[133X[101X
  
  [33X[0;0YThe  following  functions  give  access to components of wreath products and
  their elements.[133X
  
  [1X4.2-1 ComponentsOfWreathProduct[101X
  
  [33X[1;0Y[29X[2XComponentsOfWreathProduct[102X( [3XW[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a  list of two groups [K, H], where [3XW = K wr H[103X. The argument [3XW[103X must
  be a wreath product (see [14X2.1[114X).[133X
  
  [1X4.2-2 TopGroupOfWreathProduct[101X
  
  [33X[1;0Y[29X[2XTopGroupOfWreathProduct[102X( [3XW[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a group, namely the top group [22X⟨ 1_K ⟩^m × H[122X of the wreath product [22XW
  = K ≀ H[122X (see [14X2.1[114X).[133X
  
  [1X4.2-3 BaseGroupOfWreathProduct[101X
  
  [33X[1;0Y[29X[2XBaseGroupOfWreathProduct[102X( [3XW[103X[, [3Xi[103X] ) [32X function[133X
  
  [33X[0;0Yreturns a group, namely the base group [22XK^m × ⟨ 1_H[122X of the wreath product [22XW =
  K ≀ H[122X. If the optional argument [3Xi[103X is provided, the function returns the [3Xi[103X-th
  factor of the base group of [3XW[103X (see [14X2.1[114X).[133X
  
  [1X4.2-4 TopComponentOfWreathProductElement[101X
  
  [33X[1;0Y[29X[2XTopComponentOfWreathProductElement[102X( [3Xx[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a group element, namely the top component of [3Xx[103X. The argument [3Xx[103X must
  be a wreath product element (see [14X2.1[114X).[133X
  
  [1X4.2-5 BaseComponentOfWreathProductElement[101X
  
  [33X[1;0Y[29X[2XBaseComponentOfWreathProductElement[102X( [3Xx[103X[, [3Xi[103X] ) [32X function[133X
  
  [33X[0;0Yreturns  a  group  element,  namely the base component of [3Xx[103X. If the optional
  argument  [3Xi[103X  is provided, the function returns the [3Xi[103X-th base component of [3Xx[103X.
  The  argument [3Xx[103X must be a wreath product element and the optional argument [3Xi[103X
  must be an integer (see [14X2.1[114X).[133X
  
  
  [1X4.3 [33X[0;0YProperties of Wreath Product Elements[133X[101X
  
  [33X[0;0YThe  following  functions  give  access  to  important  properties of wreath
  product elements.[133X
  
  [1X4.3-1 Territory[101X
  
  [33X[1;0Y[29X[2XTerritory[102X( [3Xx[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns  a  list, namely the territory of [3Xx[103X. The argument [3Xx[103X must be a wreath
  product element (see [14X2.2[114X).[133X
  
  [1X4.3-2 IsWreathCycle[101X
  
  [33X[1;0Y[29X[2XIsWreathCycle[102X( [3Xx[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns  true  or  false.  Tests whether [3Xx[103X is a wreath cycle. The argument [3Xx[103X
  must be a wreath product element (see [14X2.2[114X).[133X
  
  [1X4.3-3 IsSparseWreathCycle[101X
  
  [33X[1;0Y[29X[2XIsSparseWreathCycle[102X( [3Xx[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns  true  or  false.  Tests  whether  [3Xx[103X  is  a sparse wreath cycle. The
  argument [3Xx[103X must be a wreath product element (see [14X2.3[114X).[133X
  
  [1X4.3-4 WreathCycleDecomposition[101X
  
  [33X[1;0Y[29X[2XWreathCycleDecomposition[102X( [3Xx[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns   a   list   containing  wreath  cycles,  namely  the  wreath  cycle
  decomposition  of  [3Xx[103X.  The  argument  [3Xx[103X  must  be  a  wreath product element
  (see [14X2.2[114X).[133X
  
  [1X4.3-5 Yade[101X
  
  [33X[1;0Y[29X[2XYade[102X( [3Xx[103X[, [3Xi[103X] ) [32X attribute[133X
  
  [33X[0;0Yreturns  a group element, namely the yade of the wreath cycle [3Xx[103X evaluated at
  the  smallest  territory  point. If the optional argument [3Xi[103X is provided, the
  function returns the yade evaluated at the point [3Xi[103X. The argument [3Xx[103X must be a
  wreath  cycle  and  the  optional  argument  [3Xi[103X  must  be an integer from the
  territory of [3Xx[103X (see [14X2.3[114X)[133X
  
  
  [1X4.4 [33X[0;0YPrinting, Viewing and Displaying[133X[101X
  
  [1X4.4-1 ViewObj[101X
  
  [33X[1;0Y[29X[2XViewObj[102X( [3Xx[103X ) [32X method[133X
  [33X[1;0Y[29X[2XPrintObj[102X( [3Xx[103X ) [32X method[133X
  [33X[1;0Y[29X[2XDisplay[102X( [3Xx[103X[, [3Xoptrec[103X] ) [32X method[133X
  
  [33X[0;0YWreath  product  elements  are  viewed,  printed  and displayed (see Section
  [14X'Reference:  View  and Print'[114X for the distinctions between these operations)
  as  generic wreath product elements (see Section [14X2.1[114X). For an example of the
  distinctions and outputs see [14X3.2[114X.[133X
  
  [33X[0;0YThe method [10XDisplay[110X allows an optional argument [3Xoptrec[103X which must be a record
  and  modifies  the  display output for the execution of a single instance of
  the command.[133X
  
  [33X[0;0YFor  modifying  the display output globally for all subsequent executions of
  [10XDisplay[110X see [2XSetDisplayOptionsForWreathProductElements[102X ([14X4.4-3[114X).[133X
  
  [33X[0;0YThe  following  components  of  [3Xoptrec[103X  are  supported.  Note,  that  in the
  following  [13Xlabels[113X  refer  to the the printing output [21X[22X1, /ldots, m[122X[121X and [21Xtop[121X as
  seen in the tutorials.[133X
  
  [8X[10Xhorizontal[110X[8X[108X
        [33X[0;6Y[9Xtrue[109X to use the horizontal printer. [13XDEFAULT[113X[133X
  
        [33X[0;6Y[9Xfalse[109X to use the vertical printer.[133X
  
  [8X[10Xlabels[110X[8X[108X
        [33X[0;6Y[9Xtrue[109X to print labels. [13XDEFAULT[113X[133X
  
        [33X[0;6Y[9Xfalse[109X to suppress labels.[133X
  
  [8X[10XlabelStyle[110X[8X[108X
        [33X[0;6Y[9X"none"[109X for labels in normal intensity. [13XDEFAULT[113X[133X
  
        [33X[0;6Y[9X"bold"[109X for labels in increased intensity.[133X
  
        [33X[0;6Y[9X"faint"[109X for labels in decreased intensity.[133X
  
  [8X[10XlabelColor[110X[8X[108X
        [33X[0;6Y[9X"default"[109X for labels in the default GAP output color. [13XDEFAULT[113X[133X
  
        [33X[0;6Y[9X"red"[109X for labels in red color.[133X
  
        [33X[0;6Y[9X"blue"[109X for labels in blue color.[133X
  
  [1X4.4-2 DisplayOptionsForWreathProductElements[101X
  
  [33X[1;0Y[29X[2XDisplayOptionsForWreathProductElements[102X(  ) [32X function[133X
  
  [33X[0;0Yprints the current global display options for wreath product elements.[133X
  
  [1X4.4-3 SetDisplayOptionsForWreathProductElements[101X
  
  [33X[1;0Y[29X[2XSetDisplayOptionsForWreathProductElements[102X( [3Xoptrec[103X ) [32X function[133X
  
  [33X[0;0Ysets the current global display options for wreath product elements.[133X
  
  [33X[0;0YThe  argument [3Xoptrec[103X must be a record with components that are valid display
  options. (see [14X4.4[114X) The components for the current global display options are
  set to the values specified by the components in [3Xoptrec[103X.[133X
  
  [1X4.4-4 ResetDisplayOptionsForWreathProductElements[101X
  
  [33X[1;0Y[29X[2XResetDisplayOptionsForWreathProductElements[102X(  ) [32X function[133X
  
  [33X[0;0Yresets  the  current  global  display options for wreath product elements to
  default.[133X
  
  
  [1X4.5 [33X[0;0YCycle Index of Wreath Products[133X[101X
  
  [33X[0;0YThe  following  functions  construct  the  cycle  index polynomial of wreath
  products in certain actions.[133X
  
  [1X4.5-1 CycleIndexWreathProductImprimitiveAction[101X
  
  [33X[1;0Y[29X[2XCycleIndexWreathProductImprimitiveAction[102X( [3XK[103X, [3XH[103X ) [32X function[133X
  
  [33X[0;0YFor  two permutation groups [3XK[103X and [3XH[103X this function constructs the cycle index
  polynomial of the wreath product [22XK ≀ H[122X in imprimitive action.[133X
  
  [33X[0;0YThe implementation is based on [P\t37].[133X
  
  [1X4.5-2 CycleIndexWreathProductProductAction[101X
  
  [33X[1;0Y[29X[2XCycleIndexWreathProductProductAction[102X( [3XK[103X, [3XH[103X ) [32X function[133X
  
  [33X[0;0YFor  two permutation groups [3XK[103X and [3XH[103X this function constructs the cycle index
  polynomial of the wreath product [22XK ≀ H[122X in product action.[133X
  
  [33X[0;0YThe implementation is based on [HH68] and [PR73].[133X
  
