Normal Subgroup of Product of Groups
Hiroyuki Okazaki ; Kenichi Arai ; Yasunari Shidama
Formalized Mathematics, Tome 19 (2011), p. 23-26 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

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.

Publié le : 2011-01-01
EUDML-ID : urn:eudml:doc:266892 
@article{bwmeta1.element.doi-10_2478_v10037-011-0004-7,
     author = {Hiroyuki Okazaki and Kenichi Arai and Yasunari Shidama},
     title = {Normal Subgroup of Product of Groups},
     journal = {Formalized Mathematics},
     volume = {19},
     year = {2011},
     pages = {23-26},
     zbl = {1276.20033},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10037-011-0004-7}
}

Hiroyuki Okazaki; Kenichi Arai; Yasunari Shidama. Normal Subgroup of Product of Groups. Formalized Mathematics, Tome 19 (2011) pp. 23-26. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10037-011-0004-7/

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. | Zbl 06213858

[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. | Zbl 1298.55008

[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.