diff --git a/docs/src/Groups/matgroup.md b/docs/src/Groups/matgroup.md
index 17ab0aa3e385..df733faa6e50 100644
--- a/docs/src/Groups/matgroup.md
+++ b/docs/src/Groups/matgroup.md
@@ -11,9 +11,10 @@ end
 matrix_group(R::Ring, m::Int, V::AbstractVector{T}; check::Bool=true) where T<:Union{MatElem,MatrixGroupElem}
 MatrixGroup{RE<:RingElem, T<:MatElem{RE}}
 MatrixGroupElem{RE<:RingElem, T<:MatElem{RE}}
-base_ring(G::MatrixGroup)
+base_ring(G::MatrixGroup{RE}) where RE <: RingElem
 degree(G::MatrixGroup)
 centralizer(G::MatrixGroup{T}, x::MatrixGroupElem{T}) where T <: FinFieldElem
+map_entries(f, G::MatrixGroup)
 ```
 
 ## Elements of matrix groups
diff --git a/src/Groups/matrices/MatGrp.jl b/src/Groups/matrices/MatGrp.jl
index 3e3ed7196569..92896f6d0339 100644
--- a/src/Groups/matrices/MatGrp.jl
+++ b/src/Groups/matrices/MatGrp.jl
@@ -102,13 +102,7 @@ function Base.deepcopy_internal(x::MatrixGroupElem, dict::IdDict)
   error("$x has neither :X nor :elm")
 end
 
-function change_base_ring(R::Ring, G::MatrixGroup)
-  g = dense_matrix_type(R)[]
-  for h in gens(G)
-    push!(g, map_entries(R, h.elm))
-  end
-  return matrix_group(g)
-end
+change_base_ring(R::Ring, G::MatrixGroup) = map_entries(R, G)
 
 ########################################################################
 #
@@ -547,6 +541,53 @@ function order(::Type{T}, G::MatrixGroup) where T <: IntegerUnion
    return T(res)::T
 end
 
+"""
+    map_entries(f, G::MatrixGroup)
+
+Return the matrix group obtained by applying `f` element-wise to
+each generator of `G`.
+
+`f` can be a ring or a field, a suitable map, or a Julia function.
+
+# Examples
+```jldoctest
+julia> mat = matrix(ZZ, 2, 2, [1, 1, 0, 1]);
+
+julia> G = matrix_group(mat);
+
+julia> G2 = map_entries(x -> -x, G)
+Matrix group of degree 2
+  over integer ring
+
+julia> is_finite(G2)
+false
+
+julia> order(map_entries(GF(3), G))
+3
+```
+"""
+function map_entries(f, G::MatrixGroup)
+  Ggens = gens(G)
+  if length(Ggens) == 0
+    z = f(zero(base_ring(G)))
+    return matrix_group(parent(z), degree(G), MatrixGroupElem[])
+  else
+    imgs = [map_entries(f, matrix(x)) for x in gens(G)]
+    return matrix_group(imgs)
+  end
+end
+
+function map_entries(R::Ring, G::MatrixGroup)
+  imgs = [map_entries(R, matrix(x)) for x in gens(G)]
+  return matrix_group(R, degree(G), imgs)
+end
+
+function map_entries(mp::Map, G::MatrixGroup)
+  imgs = [map_entries(mp, matrix(x)) for x in gens(G)]
+  return matrix_group(codomain(mp), degree(G), imgs)
+end
+
+
 ########################################################################
 #
 # Constructors
diff --git a/test/Groups/matrixgroups.jl b/test/Groups/matrixgroups.jl
index b35b6936de9b..dbba1fd97ccf 100644
--- a/test/Groups/matrixgroups.jl
+++ b/test/Groups/matrixgroups.jl
@@ -379,6 +379,38 @@ end
    @test_throws ArgumentError matrix_group([x1,x3])
 end
 
+@testset "map_entries for matrix groups" begin
+  mat = matrix(ZZ, 2, 2, [1, 1, 0, 1])
+  G = matrix_group(mat)
+  T = trivial_subgroup(G)[1]
+  @test length(gens(T)) == 0
+  for R in [GF(2), GF(3, 2), residue_ring(ZZ, 6)[1]]
+    red = map_entries(R, G)
+    @test matrix(gen(red, 1)) == map_entries(R, mat)
+    red = map_entries(R, T)
+    @test matrix(one(red)) == map_entries(R, one(mat))
+  end
+
+  F = GF(2)
+  mp = MapFromFunc(ZZ, F, x -> F(x))
+  red = map_entries(mp, G)
+  @test red == map_entries(F, G)
+  red = map_entries(mp, T)
+  @test red == map_entries(F, T)
+
+  G1 = special_linear_group(2, 9)
+  G2 = map_entries(x -> x^3, G1)
+  @test gens(G1) != gens(G2)
+  @test G1 == G2
+  T = trivial_subgroup(G1)[1]
+  @test length(gens(T)) == 0
+  @test map_entries(x -> x^3, T) == trivial_subgroup(G2)[1]
+
+  mat = matrix(QQ, 2, 2, [2, 1, 0, 1])
+  G = matrix_group(mat)
+  @test_throws ArgumentError map_entries(GF(2), G)
+end
+
 @testset "Iterator" begin
    G = SL(2,3)
    N = 0