# Normal Subgroup of Product of Groups

Hiroyuki Okazaki; Kenichi Arai; Yasunari Shidama

Formalized Mathematics (2011)

- Volume: 19, Issue: 1, page 23-26
- ISSN: 1426-2630

## Access Full Article

top## Abstract

top## How to cite

topHiroyuki Okazaki, Kenichi Arai, and Yasunari Shidama. "Normal Subgroup of Product of Groups." Formalized Mathematics 19.1 (2011): 23-26. <http://eudml.org/doc/266892>.

@article{HiroyukiOkazaki2011,

abstract = {In [6] it was formalized that the direct product of a family of groups gives a new group. In this article, we formalize that for all j ∈ I, the group G = Πi∈IGi has a normal subgroup isomorphic to Gj. Moreover, we show some relations between a family of groups and its direct product.},

author = {Hiroyuki Okazaki, Kenichi Arai, Yasunari Shidama},

journal = {Formalized Mathematics},

keywords = {direct products of groups; direct sums; normal subgroups},

language = {eng},

number = {1},

pages = {23-26},

title = {Normal Subgroup of Product of Groups},

url = {http://eudml.org/doc/266892},

volume = {19},

year = {2011},

}

TY - JOUR

AU - Hiroyuki Okazaki

AU - Kenichi Arai

AU - Yasunari Shidama

TI - Normal Subgroup of Product of Groups

JO - Formalized Mathematics

PY - 2011

VL - 19

IS - 1

SP - 23

EP - 26

AB - In [6] it was formalized that the direct product of a family of groups gives a new group. In this article, we formalize that for all j ∈ I, the group G = Πi∈IGi has a normal subgroup isomorphic to Gj. Moreover, we show some relations between a family of groups and its direct product.

LA - eng

KW - direct products of groups; direct sums; normal subgroups

UR - http://eudml.org/doc/266892

ER -

## References

top- [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. Zbl06213858
- [2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
- [3] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.
- [4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
- [5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
- [6] Artur Korniłowicz. The product of the families of the groups. Formalized Mathematics, 7(1):127-134, 1998. Zbl1298.55008
- [7] Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955-962, 1990.
- [8] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.
- [9] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5):855-864, 1990.
- [10] Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41-47, 1991.
- [11] Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573-578, 1991.
- [12] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
- [13] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.