  In the first part of this note, we introduce Tietze transformations for
$L$-presentations. These transformations enable us to generalize Tietze's
theorem for finitely presented groups to invariantly finitely $L$-presented
groups. Moreover, they allow us to prove that `being invariantly finitely
$L$-presented' is an abstract property of a group which does not depend on the
generating set.
  In the second part of this note, we consider finitely generated normal
subgroups of finitely presented groups. Benli proved that a finitely generated
normal subgroup of a finitely presented group is invariantly finitely
$L$-presented whenever its quotient is infinite cyclic. We generalize this
result to the case where the finitely presented group splits over its finitely
generated subgroup and to the case where the quotient is abelian with
torsion-free rank at most two.
