From a697884366f859b61f8752e07d5f3f71bf383a59 Mon Sep 17 00:00:00 2001
From: Max Horn <max@quendi.de>
Date: Fri, 28 Aug 2026 17:40:02 +0200
Subject: [PATCH] Fix preimage computation for field homomorphisms

PreImagesElm for field homomorphisms compared IsInjective with 1, a
GAP3 leftover: the injective branch was dead, so every nonzero element
got the empty preimage, even under bijective maps such as Frobenius
automorphisms.

With that branch reachable, PreImagesElm needs PreImagesRepresentative,
which had no applicable method for Frobenius automorphisms.  Add one
that delegates to ImagesRepresentative of the inverse; the inverse of a
Frobenius automorphism is again one (or the identity), so this cannot
recurse.

Note for #6409 (renames these operations to NC variants): the fixed
comparison then belongs in PreImagesElmNC, and the new method in
fieldfin.gi should be installed on PreImagesRepresentativeNC.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
---
 lib/field.gi                                  |  2 +-
 lib/fieldfin.gi                               |  9 ++++++++
 .../2026-08-28-PreImagesFrobenius.tst         | 23 +++++++++++++++++++
 3 files changed, 33 insertions(+), 1 deletion(-)
 create mode 100644 tst/testbugfix/2026-08-28-PreImagesFrobenius.tst

diff --git a/lib/field.gi b/lib/field.gi
index f5eadb8ede..c4ee2c4895 100644
--- a/lib/field.gi
+++ b/lib/field.gi
@@ -1322,7 +1322,7 @@ InstallMethod( PreImagesElm,
     FamRangeEqFamElm,
     [ IsFieldHomomorphism, IsObject ],
     function ( hom, elm )
-    if IsInjective( hom ) = 1 then
+    if IsInjective( hom ) then
       return [ PreImagesRepresentative( hom, elm ) ];
     elif IsZero( elm ) then
       return Source( hom );
diff --git a/lib/fieldfin.gi b/lib/fieldfin.gi
index 89b3d0aa79..612e990f38 100644
--- a/lib/fieldfin.gi
+++ b/lib/fieldfin.gi
@@ -759,6 +759,15 @@ InstallMethod( ImagesRepresentative,
     return elm ^ aut!.power;
     end );
 
+InstallMethod( PreImagesRepresentative,
+    "for Frobenius automorphism and range element",
+    FamRangeEqFamElm,
+    [ IsFrobeniusAutomorphism, IsObject ],
+    function( aut, elm )
+    # the inverse is again a Frobenius automorphism (or the identity)
+    return ImagesRepresentative( InverseGeneralMapping( aut ), elm );
+    end );
+
 InstallMethod( CompositionMapping2,
     "for two Frobenius automorphisms",
     IsIdenticalObj,
diff --git a/tst/testbugfix/2026-08-28-PreImagesFrobenius.tst b/tst/testbugfix/2026-08-28-PreImagesFrobenius.tst
new file mode 100644
index 0000000000..f6bd957556
--- /dev/null
+++ b/tst/testbugfix/2026-08-28-PreImagesFrobenius.tst
@@ -0,0 +1,23 @@
+# `PreImagesElm' for field homomorphisms compared `IsInjective' with 1
+# (a GAP3 leftover), so the injective branch was dead and every nonzero
+# element got the empty preimage. Fixing that exposed that Frobenius
+# automorphisms had no `PreImagesRepresentative' method at all.
+gap> frob := FrobeniusAutomorphism(GF(4));;
+gap> PreImagesRepresentative(frob, Z(4));
+Z(2^2)^2
+gap> ImageElm(frob, PreImagesRepresentative(frob, Z(4))) = Z(4);
+true
+gap> PreImagesElm(frob, Z(4));
+[ Z(2^2)^2 ]
+gap> PreImagesElm(frob, 0*Z(4));
+[ 0*Z(2) ]
+gap> PreImagesSet(frob, GF(4)) = GF(4);
+true
+
+# a proper power of the Frobenius automorphism over a larger field
+gap> aut := FrobeniusAutomorphism(GF(8))^2;;
+gap> x := PreImagesRepresentative(aut, Z(8));;
+gap> ImageElm(aut, x) = Z(8);
+true
+gap> PreImagesElm(aut, Z(8)) = [ x ];
+true