# Groups – Additive Notation

Formalized Mathematics (2015)

- Volume: 23, Issue: 2, page 127-160
- ISSN: 1426-2630

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topRoland Coghetto. "Groups – Additive Notation." Formalized Mathematics 23.2 (2015): 127-160. <http://eudml.org/doc/271755>.

@article{RolandCoghetto2015,

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.},

author = {Roland Coghetto},

journal = {Formalized Mathematics},

keywords = {additive group; subgroup; Lagrange theorem; conjugation; normal subgroup; index; additive topological group; basis; neighborhood; additive abelian group; Z-module; additive groups; normal subgroups; additive topological groups; bases; neighborhoods; additive Abelian groups; -modules; formalization},

language = {eng},

number = {2},

pages = {127-160},

title = {Groups – Additive Notation},

url = {http://eudml.org/doc/271755},

volume = {23},

year = {2015},

}

TY - JOUR

AU - Roland Coghetto

TI - Groups – Additive Notation

JO - Formalized Mathematics

PY - 2015

VL - 23

IS - 2

SP - 127

EP - 160

AB - 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.

LA - eng

KW - additive group; subgroup; Lagrange theorem; conjugation; normal subgroup; index; additive topological group; basis; neighborhood; additive abelian group; Z-module; additive groups; normal subgroups; additive topological groups; bases; neighborhoods; additive Abelian groups; -modules; formalization

UR - http://eudml.org/doc/271755

ER -

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