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## Formalized Mathematics

2015 | 23 | 1 | 67-73
Tytuł artykułu

### Equivalent Expressions of Direct Sum Decomposition of Groups1

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this article, the equivalent expressions of the direct sum decomposition of groups are mainly discussed. In the first section, we formalize the fact that the internal direct sum decomposition can be defined as normal subgroups and some of their properties. In the second section, we formalize an equivalent form of internal direct sum of commutative groups. In the last section, we formalize that the external direct sum leads an internal direct sum. We referred to [19], [18] [8] and [14] in the formalization.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
67-73
Opis fizyczny
Daty
wydano
2015-03-01
otrzymano
2015-02-26
online
2015-03-31
Twórcy
autor
• Shinshu University, Nagano, Japan
autor
• Shinshu University, Nagano, Japan
autor
• Shinshu University, Nagano, Japan
autor
• Shinshu University, Nagano, Japan
Bibliografia
• [1] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.
• [2] Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563–567, 1990.
• [3] Grzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213–225, 1992.
• [4] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.
• [5] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.
• [6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.
• [7] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485–492, 1996.
• [8] Nicolas Bourbaki. Elements of Mathematics. Algebra I. Chapters 1-3. Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1989.
• [9] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.
• [10] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.
• [11] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.
• [12] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.
• [13] Artur Korniłowicz. The product of the families of the groups. Formalized Mathematics, 7(1):127–134, 1998.
• [14] Serge Lang. Algebra. Springer, 3rd edition, 2005.
• [15] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1): 103–108, 1993.
• [16] Kazuhisa Nakasho, Hiroshi Yamazaki, Hiroyuki Okazaki, and Yasunari Shidama. Definition and properties of direct sum decomposition of groups. Formalized Mathematics, 23 (1):15–27, 2015. doi:10.2478/forma-2015-0002.
• [17] Hiroyuki Okazaki, Kenichi Arai, and Yasunari Shidama. Normal subgroup of product of groups. Formalized Mathematics, 19(1):23–26, 2011. doi:10.2478/v10037-011-0004-7.
• [18] D. Robinson. A Course in the Theory of Groups. Springer New York, 2012.
• [19] J.J. Rotman. An Introduction to the Theory of Groups. Springer, 1995.
• [20] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115–122, 1990.
• [21] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.
• [22] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.
• [23] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5): 855–864, 1990.
• [24] Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955–962, 1990.
• [25] Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41–47, 1991.
• [26] Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573–578, 1991.
• [27] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.
• [28] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.
• [29] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.
Typ dokumentu
Bibliografia
Identyfikatory
bwmeta1.id-class.MML
GROUP _20