diff --git a/gap/bytable.gi b/gap/bytable.gi
index a1fb785..f57efee 100644
--- a/gap/bytable.gi
+++ b/gap/bytable.gi
@@ -24,7 +24,31 @@ InstallMethod(IsElementwiseIdempotentTable, "for matrix",
   T -> ForAll([1..Length(T)], i->(T[i,i]=i))
 );
 
+## And a general principle: collections from finite families are finite.
+
+InstallMethod(IsFinite, "for any collection (with a finite element family)",
+  [IsCollection],
+  function(C)            
+    local ef;
+    ef := ElementsFamily(FamilyObj(C));
+    if HasIsFinite(ef) and IsFinite(ef) then return true; fi;
+    TryNextMethod();
+    return fail;
+end);
+
+      
 ## And now create them from multiplication tables
+
+#First a helper function
+FiniteMagmaCreator@ := function(tbl, const, filts)
+  local M;
+  M := MagmaByMultiplicationTableCreatorNC(
+         T, const, filts and IsMagmaByMultiplicationTableObj);
+  SetIsFinite(ElementsFamily(FamilyObject(M)));
+  return M;
+end;
+
+
 InstallGlobalFunction(LeftQuasigroupByMultiplicationTable,
   function(T)
     if not IsLeftQuasigroupTable(T) then
@@ -36,8 +60,7 @@ InstallGlobalFunction(LeftQuasigroupByMultiplicationTable,
 end);
 
 InstallGlobalFunction(LeftQuasigroupByMultiplicationTableNC,
-  T -> MagmaByMultiplicationTableCreatorNC(T, LeftQuasigroupNC,
-         IsLeftQuotientElement and IsMagmaByMultiplicationTableObj)
+  T -> FiniteMagmaCreator@(T, LeftQuasigroupNC, IsLeftQuotientElement)
 );
 
 InstallGlobalFunction(RightQuasigroupByMultiplicationTable,
@@ -50,8 +73,7 @@ InstallGlobalFunction(RightQuasigroupByMultiplicationTable,
 end);
 
 InstallGlobalFunction(RightQuasigroupByMultiplicationTableNC,
-  T -> MagmaByMultiplicationTableCreatorNC(T, RightQuasigroupNC,
-         IsRightQuotientElement and IsMagmaByMultiplicationTableObj)
+  T -> FiniteMagmaCreator@(T, RightQuasigroupNC, IsRightQuotientElement)
 );
 
 InstallGlobalFunction(LeftRackByMultiplicationTable,
@@ -68,10 +90,8 @@ InstallGlobalFunction(LeftRackByMultiplicationTable,
   end);
 
 InstallGlobalFunction(LeftRackByMultiplicationTableNC,
-  T -> MagmaByMultiplicationTableCreatorNC(T, LeftRackNC,
-         IsLeftQuotientElement and IsLSelfDistElement and 
-         IsMagmaByMultiplicationTableObj
-       )
+  T -> FiniteMagmaCreator@(T, LeftRackNC,
+                           IsLeftQuotientElement and IsLSelfDistElement)
 );
 
 InstallGlobalFunction(LeftQuandleByMultiplicationTable,
@@ -89,10 +109,8 @@ InstallGlobalFunction(LeftQuandleByMultiplicationTable,
   end);
 
 InstallGlobalFunction(LeftQuandleByMultiplicationTableNC,
-  T -> MagmaByMultiplicationTableCreatorNC(T, LeftQuandleNC,
-         IsLeftQuotientElement and IsLSelfDistElement and IsIdempotent and
-         IsMagmaByMultiplicationTableObj
-       )
+  T -> FiniteMagmaCreator@(T, LeftQuandleNC,
+         IsLeftQuotientElement and IsLSelfDistElement and IsIdempotent)
 );
 
 InstallGlobalFunction(RightRackByMultiplicationTable,
@@ -109,10 +127,8 @@ InstallGlobalFunction(RightRackByMultiplicationTable,
 end);
 
 InstallGlobalFunction(RightRackByMultiplicationTableNC,
-  T -> MagmaByMultiplicationTableCreatorNC(T, RightRackNC,
-         IsRightQuotientElement and IsRSelfDistElement and 
-         IsMagmaByMultiplicationTableObj
-       )
+  T -> FiniteMagmaCreator@(T, RightRackNC,
+         IsRightQuotientElement and IsRSelfDistElement)
 );
 
 InstallGlobalFunction(RightQuandleByMultiplicationTable,
@@ -130,10 +146,8 @@ InstallGlobalFunction(RightQuandleByMultiplicationTable,
 end);
 
 InstallGlobalFunction(RightQuandleByMultiplicationTableNC,
-  T -> MagmaByMultiplicationTableCreatorNC(T, RightQuandleNC,
-         IsRightQuotientElement and IsRSelfDistElement and IsIdempotent and
-         IsMagmaByMultiplicationTableObj
-       )
+  T -> FiniteMagmaCreator@(T, RightQuandleNC,
+         IsRightQuotientElement and IsRSelfDistElement and IsIdempotent)
 );
 
 ## And define the operations