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A Note on Invariantly Finitely $L$-Presented Groups

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.