WIP: Add FindImageSnSmallDegree

gap/SnAnUnknownDegree.gi

@@ -665,6 +665,28 @@ RECOG.StandardGenerators := function(ri, g, c, k, eps, N)
     fi;
 end;
 
+# In Conder's Thesis: Algorithm 7, ConjugateMap
+# returns an element c such that i^c = j.
+RECOG.FindAnElementMappingIToJ := function(s, t, i, j)
+    if i < 3 then
+        if j < 3 then
+            return t^(j-i);
+        else
+            return t^(3-i)*s^(j-3);
+        fi;
+    else
+        return s^(j-i);
+    fi;
+end;
+
+# TODO: Finish Algorithm
+# In Conder's Thesis: Algorithm 10, ElementToSmallDegreePermutation
+# Note that the arguments are in different order, so they are more consistent with the GAP function FindImageSn.
+# returns image of g under the isomorphism
+RECOG.FindImageSnSmallDegree := function(ri, n, g, s, t, E)
+
+end;
+
 # This function is an excerpt of the function RECOG.RecogniseSnAn in gap/SnAnBB.gi
 # ri : recog info record with group G,
 # n : degree
@@ -681,16 +703,26 @@ end;
 # - slpToStdGens: an SLP to stdGens.
 #
 # TODO: Image Computation requires n >= 11.
+# TODO: IsoData for n < 11
 RECOG.ConstructSnAnIsomorphism := function(ri, n, stdGensAnWithMemory)
     local grp, stdGensAn, xis, gImage, foundOddPermutation, slp, eval,
         hWithMemory, bWithMemory, stdGensSnWithMemory, b, h, g;
     grp := GroupWithMemory(Grp(ri));
     stdGensAn := StripMemory(stdGensAnWithMemory);
-    xis := RECOG.ConstructXiAn(n, stdGensAn[1], stdGensAn[2]);
+    if n < 11 then
+        E := [stdGensAn[2], (~[1]^s)^(2*(n mod 2) - 1), (~[2]^s)^(2*(n mod 2) - 1)];
+    else
+        xis := RECOG.ConstructXiAn(n, stdGensAn[1], stdGensAn[2]);
+    fi;
     for g in GeneratorsOfGroup(grp) do
-        gImage := RECOG.FindImageAn(ri, n, StripMemory(g),
-                                    stdGensAn[1], stdGensAn[2],
-                                    xis[1], xis[2]);
+        if n < 11 then
+            gImage := RECOG.FindImageSnSmallDegree(ri, n, StripMemory(g),
+                                                   stdGensAn[1], stdGensAn[2], E);
+        else
+            gImage := RECOG.FindImageAn(ri, n, StripMemory(g),
+                                        stdGensAn[1], stdGensAn[2],
+                                        xis[1], xis[2]);
+        fi;
         if gImage = fail then return fail; fi;
         if SignPerm(gImage) = -1 then
             # we found an odd permutation,
@@ -724,11 +756,18 @@ RECOG.ConstructSnAnIsomorphism := function(ri, n, stdGensAnWithMemory)
     if not RECOG.SatisfiesSnPresentation(ri, n, b, h) then
         return fail;
     fi;
-    xis := RECOG.ConstructXiSn(n, b, h);
+    if n >= 11 then
+        xis := RECOG.ConstructXiSn(n, b, h);
+    fi;
     for g in GeneratorsOfGroup(grp) do
-        gImage := RECOG.FindImageSn(ri, n, StripMemory(g),
-                                    b, h,
-                                    xis[1], xis[2]);
+        if n < 11 then
+            gImage := RECOG.FindImageSnSmallDegree(ri, n, StripMemory(g),
+                                                   stdGensAn[1], stdGensAn[2], E);
+        else
+            gImage := RECOG.FindImageSn(ri, n, StripMemory(g),
+                                        b, h,
+                                        xis[1], xis[2]);
+        fi;
         if gImage = fail then return fail; fi;
         slp := RECOG.SLPforSn(n, gImage);
         eval := ResultOfStraightLineProgram(slp, [h, b]);