From 2521a38635179e8078ec19f015896d2a20946cba Mon Sep 17 00:00:00 2001
From: Glen Whitney <gwhitney@post.harvard.edu>
Date: Tue, 24 Oct 2017 16:16:27 +0200
Subject: [PATCH] Generalize CayleyTableByPerms so that it is more useful in
 rack/quandle context

---
 gap/quasigroups.gd |  2 +-
 gap/quasigroups.gi | 40 +++++++++++++++++++++++++++-------------
 2 files changed, 28 insertions(+), 14 deletions(-)

diff --git a/gap/quasigroups.gd b/gap/quasigroups.gd
index 27ee79a..e874d7b 100644
--- a/gap/quasigroups.gd
+++ b/gap/quasigroups.gd
@@ -117,7 +117,7 @@ DeclareOperation( "LoopFromFile", [ IsString, IsString ] );
 ##  CREATING QUASIGROUPS AND LOOPS BY SECTIONS
 ##  -------------------------------------------------------------------------
 
-DeclareOperation( "CayleyTableByPerms", [ IsPermCollection ] );
+DeclareGlobalFunction("CayleyTableByPerms");
 DeclareOperation( "QuasigroupByLeftSection", [ IsPermCollection ] );
 DeclareOperation( "LoopByLeftSection", [ IsPermCollection ] );
 DeclareOperation( "QuasigroupByRightSection", [ IsPermCollection ] );
diff --git a/gap/quasigroups.gi b/gap/quasigroups.gi
index f691a0a..3fa9c69 100644
--- a/gap/quasigroups.gi
+++ b/gap/quasigroups.gi
@@ -438,30 +438,44 @@ end );
 
 #############################################################################
 ##
-#O  CayleyTableByPerms( perms )
+#O  CayleyTableByPerms( perms, [X] )
 ##
-##  Given a set <perms> of n permutations of an n-element set X, returns
-##  n by n Cayley table ct such that ct[i][j] = X[j]^perms[i].
-##  The operation is safe only if at most one permutation of <perms> is
-##  the identity permutation, and all other permutations of <perms>
-##  move all points of X.
+##  Given a set <perms> of n permutations of an n-element set <X> of natural
+##  numbers, returns an n by n Cayley table ct such that 
+##    ct[i][j] = X[j]^perms[i]. 
+##
+##  Note that the argument <X> is optional, and if omitted, the function will
+##  assume that at most one permutation of <perms> is the identity
+##  permutation, and that all other permutations of <perms> 
+##  move all points of <X>.
 
-InstallMethod( CayleyTableByPerms,
-    "for a list of permutations",
-    [ IsPermCollection ],
-function( perms )
+InstallGlobalFunction( CayleyTableByPerms,
+function( perms, rest... )
     local n, pts, max;
     n := Length( perms );
     if n=1 then
         return [ [ 1 ] ];
     fi;
+    
     # one of perms[ 1 ], perms[ 2 ] must move all points
-    pts := MovedPoints( perms[ 2 ] );
-    if pts = [] then
+    if Length(rest) > 0 then
+      pts := rest[0];
+    else  
+      pts := MovedPoints( perms[ 2 ] );
+      if pts = [] then
         pts := MovedPoints( perms[ 1 ] );
+      fi;
+    fi;
+    if Length(pts) <> n then
+      Error("perms for cayley table of size ", n, " cannot act on ", 
+            Length(pts), " points.");
     fi;
     max := Maximum( pts );
-    # we permute the whole interval [1..max] and then keep only those coordinates corresponding to pts
+    if max = n then
+      # Common case, we are permuting [1..n]
+      return List( perms, x -> ListPerm(x, n));
+    fi;
+    # Otherwise we permute the whole interval [1..max] and then keep only those coordinates corresponding to pts
     return List( perms, p -> Permuted( [1..max], p^(-1) ){ pts } );
 end);