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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year


SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

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1898-9934
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Volume 23, Issue 1 (Mar 2015)

Issues

Definition and Properties of Direct Sum Decomposition of Groups1

Kazuhisa Nakasho
  • Shinshu University, Nagano, Japan
/ Hiroshi Yamazaki
  • Shinshu University, Nagano, Japan
/ Hiroyuki Okazaki
  • Shinshu University, Nagano, Japan
/ Yasunari Shidama
  • Shinshu University, Nagano, Japan
Published Online: 2015-03-31 | DOI: https://doi.org/10.2478/forma-2015-0002

Summary

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.

MSC: 20E34; 03B35

Keywords: group theory; direct sum decomposition

MML: identifier: GROUP_19; version: 8.1.03 5.29.1227

References

  • [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Google Scholar

  • [2] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.Google Scholar

  • [3] Grzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213–225, 1992.Google Scholar

  • [4] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Google Scholar

  • [5] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Google Scholar

  • [6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Google Scholar

  • [7] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485–492, 1996.Google Scholar

  • [8] Nicolas Bourbaki. Elements of Mathematics. Algebra I. Chapters 1-3. Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1989.Google Scholar

  • [9] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.Google Scholar

  • [10] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Google Scholar

  • [11] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Google Scholar

  • [12] Czesław Byliński. Basic functions and operations on functions. Formalized Mathematics, 1(1):245–254, 1990.Google Scholar

  • [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.Google Scholar

  • [14] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Google Scholar

  • [15] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Google Scholar

  • [16] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Google Scholar

  • [17] Artur Korniłowicz. The product of the families of the groups. Formalized Mathematics, 7(1):127–134, 1998.Google Scholar

  • [18] Serge Lang. Algebra. Springer, 3rd edition, 2005.Google Scholar

  • [19] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1): 103–108, 1993.Google Scholar

  • [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.CrossrefGoogle Scholar

  • [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.CrossrefGoogle Scholar

  • [22] D. Robinson. A Course in the Theory of Groups. Springer New York, 2012.Google Scholar

  • [23] J.J. Rotman. An Introduction to the Theory of Groups. Springer, 1995.Google Scholar

  • [24] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329–334, 1990.Google Scholar

  • [25] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.Google Scholar

  • [26] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.Google Scholar

  • [27] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.Google Scholar

  • [28] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5): 855–864, 1990.Google Scholar

  • [29] Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41–47, 1991.Google Scholar

  • [30] Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573–578, 1991.Google Scholar

  • [31] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Google Scholar

  • [32] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Google Scholar

  • [33] Katarzyna Zawadzka. Solvable groups. Formalized Mathematics, 5(1):145–147, 1996.Google Scholar

About the article

Received: 2014-12-31

Published Online: 2015-03-31

Published in Print: 2015-03-01


1This work was supported by JSPS KAKENHI 22300285.


Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, DOI: https://doi.org/10.2478/forma-2015-0002.

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© Kazuhisa Nakasho et al.. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. BY-SA 3.0

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