We translate the articles covering group theory already available in the Mizar Mathematical Library from multiplicative into additive notation. We adapt the works of Wojciech A. Trybulec [41, 42, 43] and Artur Korniłowicz .
In particular, these authors have defined the notions of group, abelian group, power of an element of a group, order of a group and order of an element, subgroup, coset of a subgroup, index of a subgroup, conjugation, normal subgroup, topological group, dense subset and basis of a topological group. Lagrange’s theorem and some other theorems concerning these notions [9, 24, 22] are presented.
Note that “The term ℤ-module is simply another name for an additive abelian group” . We take an approach different than that used by Futa et al.  to use in a future article the results obtained by Artur Korniłowicz . Indeed, Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [23, 10]. Our goal is to define the convergence of a sequence and the convergence of a series in an abelian topological group  using the notion of filters.
 Edwin Hewitt and Kenneth A. Ross. Abstract Harmonic Analysis: Volume I. Structure of Topological Groups. Integration. Theory Group Representations volume 115. Springer Science and Business Media 2012.
 Johannes Hölzl Fabian Immler and Brian Huffman. Type classes and filters for mathematical analysis in Isabelle/HOL. In Interactive Theorem Proving pages 279-294. Springer 2013.
 Teturo Inui Yukito Tanabe and Yositaka Onodera. Group theory and its applications in physics volume 78. Springer Science and Business Media 2012.
 Artur Korniłowicz. The definition and basic properties of topological groups. Formalized Mathematics 7(2):217-225 1998.
 Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics 1(5):829-832 1990.
 Christopher Norman. Basic theory of additive Abelian groups. In Finitely Generated Abelian Groups and Similarity of Matrices over a Field Springer Undergraduate Mathematics Series pages 47-96. Springer 2012. ISBN 978-1-4471-2729-1. doi:10.1007/978-1-4471-2730-7 2.