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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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Volume 23, Issue 2


Groups – Additive Notation

Roland Coghetto
Published Online: 2015-08-13 | DOI: https://doi.org/10.1515/forma-2015-0013


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.

MSC: 20A05; 20K00; 03B35

Keywords: additive group; subgroup; Lagrange theorem; conjugation; normal subgroup; index; additive topological group; basis; neighborhood; additive abelian group; Z-module

MML: identifier: GROUP_1A; version:


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About the article

Received: 2015-04-30

Published Online: 2015-08-13

Published in Print: 2015-06-01

Citation Information: Formalized Mathematics, Volume 23, Issue 2, Pages 127–160, ISSN (Online) 1898-9934, DOI: https://doi.org/10.1515/forma-2015-0013.

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© by Roland Coghetto. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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