## Abstract

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 [25].

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” [27]. We take an approach different than that used by Futa et al. [21] to use in a future article the results obtained by Artur Korniłowicz [25]. 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 [11] using the notion of filters.

## References

[1] Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565-582, 2001.

[2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.

[3] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543-547, 1990.

[4] Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563-567, 1990.

[5] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.

[6] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.

[7] Grzegorz Bancerek. Zermelo theorem and axiom of choice. Formalized Mathematics, 1 (2):265-267, 1990.

[8] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433-439, 1990.

[9] Richard E. Blahut. Cryptography and Secure Communication. Cambridge University Press, 2014.

[10] Sylvie Boldo, Catherine Lelay, and Guillaume Melquiond. Formalization of real analysis: A survey of proof assistants and libraries. Mathematical Structures in Computer Science, pages 1-38, 2014.

[11] Nicolas Bourbaki. General Topology: Chapters 1-4. Springer Science and Business Media, 2013.

[12] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.

[13] Czesław Bylinski. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643-649, 1990.

[14] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.

[15] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.

[16] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.

[17] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.

[18] Agata Darmochwał. Compact spaces. Formalized Mathematics, 1(2):383-386, 1990.

[19] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.

[20] Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990.

[21] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Z-modules. Formalized Mathematics, 20(1):47-59, 2012. doi:10.2478/v10037-012-0007-z.

[22] 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.

[23] 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.

[24] Teturo Inui, Yukito Tanabe, and Yositaka Onodera. Group theory and its applications in physics, volume 78. Springer Science and Business Media, 2012.

[25] Artur Korniłowicz. The definition and basic properties of topological groups. Formalized Mathematics, 7(2):217-225, 1998.

[26] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.

[27] 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.

[28] Beata Padlewska. Locally connected spaces. Formalized Mathematics, 2(1):93-96, 1991.

[29] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.

[30] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.

[31] Alexander Yu. Shibakov and Andrzej Trybulec. The Cantor set. Formalized Mathematics, 5(2):233-236, 1996.

[32] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4): 535-545, 1991.

[33] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990.

[34] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.

[35] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.

[36] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.

[37] Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1 (2):369-376, 1990.

[38] Andrzej Trybulec. Baire spaces, Sober spaces. Formalized Mathematics, 6(2):289-294, 1997.

[39] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1 (1):187-190, 1990.

[40] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.

[41] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.

[42] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5): 855-864, 1990.

[43] Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955-962, 1990.

[44] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.

[45] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

[46] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.

[47] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

[48] Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231-237, 1990.