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## Formalized Mathematics

2015 | 23 | 1 | 15-27
Tytuł artykułu

### Definition and Properties of Direct Sum Decomposition of Groups1

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
15-27
Opis fizyczny
Daty
wydano
2015-03-01
otrzymano
2014-12-31
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. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.
• [2] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 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. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.
• [10] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.
• [11] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.
• [12] Czesław Byliński. Basic functions and operations on functions. Formalized Mathematics, 1(1):245–254, 1990.
• [13] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.
• [14] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.
• [15] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.
• [16] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.
• [17] Artur Korniłowicz. The product of the families of the groups. Formalized Mathematics, 7(1):127–134, 1998.
• [18] Serge Lang. Algebra. Springer, 3rd edition, 2005.
• [19] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1): 103–108, 1993.
• [20] 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.
• [21] Hiroyuki Okazaki, Hiroshi Yamazaki, and Yasunari Shidama. Isomorphisms of direct products of finite commutative groups. Formalized Mathematics, 21(1):65–74, 2013. doi:10.2478/forma-2013-0007.
• [22] D. Robinson. A Course in the Theory of Groups. Springer New York, 2012.
• [23] J.J. Rotman. An Introduction to the Theory of Groups. Springer, 1995.
• [24] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329–334, 1990.
• [25] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.
• [26] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.
• [27] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.
• [28] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5): 855–864, 1990.
• [29] Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41–47, 1991.
• [30] Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573–578, 1991.
• [31] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.
• [32] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.
• [33] Katarzyna Zawadzka. Solvable groups. Formalized Mathematics, 5(1):145–147, 1996.
Typ dokumentu
Bibliografia
Identyfikatory
bwmeta1.id-class.MML
GROUP_19