From 753af27ffc5bbe7966cc21fa965b161e40a9a09b Mon Sep 17 00:00:00 2001
From: Tommy Hofmann <thofma@gmail.com>
Date: Fri, 9 Jun 2023 22:43:56 +0200
Subject: [PATCH] Add absolute_automorphism_group(::Type{PermGroup}, ...)

---
 experimental/GModule/GaloisCohomology.jl |  5 +++++
 test/NumberTheory/nmbthy.jl              | 11 +++++++++++
 2 files changed, 16 insertions(+)

diff --git a/experimental/GModule/GaloisCohomology.jl b/experimental/GModule/GaloisCohomology.jl
index 0e52f8c8af22..9eef6fc01af3 100644
--- a/experimental/GModule/GaloisCohomology.jl
+++ b/experimental/GModule/GaloisCohomology.jl
@@ -56,6 +56,11 @@ function Oscar.automorphism_group(::Type{PermGroup}, K, k)
   return codomain(mH), mmH
 end
 
+function Oscar.absolute_automorphism_group(::Type{PermGroup}, k)
+  G, mG = absolute_automorphism_group(k)
+  mH = isomorphism(PermGroup, G)
+  return codomain(mH), inv(mH)*mG
+end
 
 """
 The natural `ZZ[H]` module where `H`, a subgroup of the
diff --git a/test/NumberTheory/nmbthy.jl b/test/NumberTheory/nmbthy.jl
index 1f081a860b76..f882f3a9d4a9 100644
--- a/test/NumberTheory/nmbthy.jl
+++ b/test/NumberTheory/nmbthy.jl
@@ -30,3 +30,14 @@ end
   F2 = GF(3,5);
   @test_throws AssertionError disc_log(gen(F), gen(F2))
 end
+
+begin
+  Qx, x = QQ["x"]
+  k, a = number_field(x^2 - 18, "a")
+  kt, t = k["t"];
+  K, b = number_field(t^4 + (a + 6)*t^2 + 2a + 9, "b")
+  G, m = automorphism_group(PermGroup, K)
+  h = m(one(G))
+  @test h(b) == b && h(K(a)) == K(a)
+  @test order(G) == 4 && is_cyclic(G)
+end