# Definition and Properties of Direct Sum Decomposition of Groups1

Formalized Mathematics (2015)

• Volume: 23, Issue: 1, page 15-27
• ISSN: 1426-2630

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## Abstract

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In this article, direct sum decomposition of group is mainly discussed. In the second section, support of element of direct product group is defined and its properties are formalized. It is formalized here that an element of direct product group belongs to its direct sum if and only if support of the element is finite. In the third section, product map and sum map are prepared. In the fourth section, internal and external direct sum are defined. In the last section, an equivalent form of internal direct sum is proved. We referred to [23], [22], [8] and [18] in the formalization.

## How to cite

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Kazuhisa Nakasho, et al. "Definition and Properties of Direct Sum Decomposition of Groups1." Formalized Mathematics 23.1 (2015): 15-27. <http://eudml.org/doc/270933>.

@article{KazuhisaNakasho2015,
abstract = {In this article, direct sum decomposition of group is mainly discussed. In the second section, support of element of direct product group is defined and its properties are formalized. It is formalized here that an element of direct product group belongs to its direct sum if and only if support of the element is finite. In the third section, product map and sum map are prepared. In the fourth section, internal and external direct sum are defined. In the last section, an equivalent form of internal direct sum is proved. We referred to [23], [22], [8] and [18] in the formalization.},
author = {Kazuhisa Nakasho, Hiroshi Yamazaki, Hiroyuki Okazaki, Yasunari Shidama},
journal = {Formalized Mathematics},
keywords = {group theory; direct sum decomposition; direct sums of groups; direct products; direct sum decompositions},
language = {eng},
number = {1},
pages = {15-27},
title = {Definition and Properties of Direct Sum Decomposition of Groups1},
url = {http://eudml.org/doc/270933},
volume = {23},
year = {2015},
}

TY - JOUR
AU - Kazuhisa Nakasho
AU - Hiroshi Yamazaki
AU - Hiroyuki Okazaki
AU - Yasunari Shidama
TI - Definition and Properties of Direct Sum Decomposition of Groups1
JO - Formalized Mathematics
PY - 2015
VL - 23
IS - 1
SP - 15
EP - 27
AB - In this article, direct sum decomposition of group is mainly discussed. In the second section, support of element of direct product group is defined and its properties are formalized. It is formalized here that an element of direct product group belongs to its direct sum if and only if support of the element is finite. In the third section, product map and sum map are prepared. In the fourth section, internal and external direct sum are defined. In the last section, an equivalent form of internal direct sum is proved. We referred to [23], [22], [8] and [18] in the formalization.
LA - eng
KW - group theory; direct sum decomposition; direct sums of groups; direct products; direct sum decompositions
UR - http://eudml.org/doc/270933
ER -

## References

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