Remove testMembership, use testDecomposition instead.

Adjust documentation of `ListWreathProductElement`.

It is often sufficient to have elements in the parent
wreath product `Sym(l) \wr Sym(m)` or `GL(n,q) \wr Sym(m)`.
For this reason the `testMembership` option
is replaced by the `testDecomposition` option,
which just tests for membership in the parent wreath product.

lib/gprd.gd

@@ -574,27 +574,30 @@ DeclareRepresentation("IsWreathProductElementDefaultRep",
 ##
 ##  <#GAPDoc Label="ListWreathProductElement">
 ##  <ManSection>
-##  <Func Name="ListWreathProductElement" Arg='G, x[, testMembership]'/>
+##  <Func Name="ListWreathProductElement" Arg='G, x[, testDecomposition]'/>
 ##  <Oper Name="ListWreathProductElementNC" Arg='G, x, testDecomposition'/>
 ##
 ##  <Description>
 ##  Let <A>x</A> be an element of a wreath product <A>G</A>
 ##  where <M>G = K \wr H</M> and <M>H</M> acts
 ##  as a finite permutation group of degree <M>m</M>.
-##  We can identify the element <A>x</A> with a tuple <M>(f_1, \dots, f_m; h)</M>,
+##  We can identify the element <A>x</A> with a tuple <M>(f_1, \ldots, f_m; h)</M>,
 ##  where <M>f_i \in K</M> is the <M>i</M>-th base component of <A>x</A>
 ##  and <M>h \in H</M> is the top component of <A>x</A>.
 ##  <P/>
-##  <Ref Func="ListWreathProductElement"/> returns a list <M>[f_1, \dots, f_m, h]</M>
-##  containing the components of <A>x</A>.
+##  <Ref Func="ListWreathProductElement"/> returns a list <M>[f_1, \ldots, f_m, h]</M>
+##  containing the components of <A>x</A> or <K>fail</K> if <A>x</A> cannot be decomposed in the wreath product.
 ##  <P/>
-##  If ommited, the argument <A>testMembership</A> defaults to true.
+##  If ommited, the argument <A>testDecomposition</A> defaults to true.
+##  If <A>testDecomposition</A> is true, <Ref Func="ListWreathProductElement"/> makes additional tests to ensure
+##  that the computed decomposition of <A>x</A> is correct,
+##  i.e. it checks that <A>x</A> is an element of the parent wreath product of <A>G</A>:
 ##  <P/>
-##  If <A>testMembership</A> is true, <Ref Func="ListWreathProductElement"/> returns fail
-##  if <A>x</A> is not an element of <A>G</A>.
+##  If <M>K \leq \mathop{Sym}(l)</M>, this ensures that <M>x \in \mathop{Sym}(l) \wr \mathop{Sym}(m)</M>
+##  where the parent wreath product is considered in the same action as <A>G</A>,
+##  i.e. either in imprimitive action or product action.
 ##  <P/>
-##  If <A>testDecomposition</A> is true, <Ref Func="ListWreathProductElementNC"/> makes additional tests to ensure that
-##  the decomposition of <A>x</A> can be used for testing membership in <A>G</A>.
+##  If <M>K \leq \mathop{GL}(n,q)</M>, this ensures that <M>x \in \mathop{GL}(n,q) \wr \mathop{Sym}(m)</M>.
 ##  </Description>
 ##  </ManSection>
 ##  <#/GAPDoc>
@@ -613,13 +616,13 @@ DeclareOperation( "ListWreathProductElementNC", [HasWreathProductInfo, IsObject,
 ##  <Oper Name="WreathProductElementListNC" Arg='G, list'/>
 ##
 ##  <Description>
-##  Let <A>list</A> be equal to <M>[f_1, \dots, f_m, h]</M> and <A>G</A> be a wreath product
+##  Let <A>list</A> be equal to <M>[f_1, \ldots, f_m, h]</M> and <A>G</A> be a wreath product
 ##  where <M>G = K \wr H</M>, <M>H</M> acts
 ##  as a finite permutation group of degree <M>m</M>,
 ##  <M>f_i \in K</M> and <M>h \in H</M>.
 ##  <P/>
 ##  <Ref Func="WreathProductElementList"/> returns the element <M>x \in G</M>
-##  identified by the tuple <M>(f_1, \dots, f_m; h)</M>.
+##  identified by the tuple <M>(f_1, \ldots, f_m; h)</M>.
 ##  </Description>
 ##  </ManSection>
 ##  <#/GAPDoc>

lib/gprd.gi

@@ -901,34 +901,19 @@ end);
 #M  ListWreathProductElement(<G>, <x>[, <testMembership>])
 ##
 InstallGlobalFunction( ListWreathProductElement,
-function(G, x, testMembership...)
-  local info, list, i;
-  if Length(testMembership) = 0 then
-    testMembership := true;
-  elif Length(testMembership) = 1 then
-    testMembership := testMembership[1];
-  elif Length(testMembership) > 1 then
+function(G, x, testDecomposition...)
+  local info;
+  if Length(testDecomposition) = 0 then
+    testDecomposition := true;
+  elif Length(testDecomposition) = 1 then
+    testDecomposition := testDecomposition[1];
+  elif Length(testDecomposition) > 1 then
     ErrorNoReturn("too many arguments");
   fi;
   if not HasWreathProductInfo(G) then
     ErrorNoReturn("usage: <G> must be a wreath product");
   fi;
-  info := WreathProductInfo(G);
-  list := ListWreathProductElementNC(G, x, testMembership);
-  if testMembership then
-    if list = fail then
-      return fail;
-    fi;
-    for i in [1 .. info.degI] do
-      if not list[i] in info.groups[1] then
-        return fail;
-      fi;
-    od;
-    if not list[info.degI + 1] in info.groups[2] then
-      return fail;
-    fi;
-  fi;
-  return list;
+  return ListWreathProductElementNC(G, x, testDecomposition);
 end);
 
 InstallMethod( ListWreathProductElementNC, "generic wreath product", true,