Definition and Properties of Direct Sum Decomposition of Groups1

Kazuhisa Nakasho; Hiroshi Yamazaki; Hiroyuki Okazaki; Yasunari Shidama

Formalized Mathematics (2015)

  • Volume: 23, Issue: 1, page 15-27
  • ISSN: 1426-2630

Abstract

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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.

How to cite

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Kazuhisa Nakasho, et al. "Definition and Properties of Direct Sum Decomposition of Groups1." Formalized Mathematics 23.1 (2015): 15-27. <http://eudml.org/doc/270933>.

@article{KazuhisaNakasho2015,
abstract = {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.},
author = {Kazuhisa Nakasho, Hiroshi Yamazaki, Hiroyuki Okazaki, Yasunari Shidama},
journal = {Formalized Mathematics},
keywords = {group theory; direct sum decomposition; direct sums of groups; direct products; direct sum decompositions},
language = {eng},
number = {1},
pages = {15-27},
title = {Definition and Properties of Direct Sum Decomposition of Groups1},
url = {http://eudml.org/doc/270933},
volume = {23},
year = {2015},
}

TY - JOUR
AU - Kazuhisa Nakasho
AU - Hiroshi Yamazaki
AU - Hiroyuki Okazaki
AU - Yasunari Shidama
TI - Definition and Properties of Direct Sum Decomposition of Groups1
JO - Formalized Mathematics
PY - 2015
VL - 23
IS - 1
SP - 15
EP - 27
AB - 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.
LA - eng
KW - group theory; direct sum decomposition; direct sums of groups; direct products; direct sum decompositions
UR - http://eudml.org/doc/270933
ER -

References

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  1. [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990. 
  2. [2] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990. 
  3. [3] Grzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213–225, 1992. 
  4. [4] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990. Zbl06213858
  5. [5] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990. 
  6. [6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990. 
  7. [7] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485–492, 1996. 
  8. [8] Nicolas Bourbaki. Elements of Mathematics. Algebra I. Chapters 1-3. Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1989. Zbl0673.00001
  9. [9] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990. 
  10. [10] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990. 
  11. [11] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990. 
  12. [12] Czesław Byliński. Basic functions and operations on functions. Formalized Mathematics, 1(1):245–254, 1990. 
  13. [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. [14] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990. 
  15. [15] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990. 
  16. [16] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990. 
  17. [17] Artur Korniłowicz. The product of the families of the groups. Formalized Mathematics, 7(1):127–134, 1998. Zbl1298.55008
  18. [18] Serge Lang. Algebra. Springer, 3rd edition, 2005. 
  19. [19] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1): 103–108, 1993. 
  20. [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. Zbl1276.20033
  21. [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. Zbl1277.20068
  22. [22] D. Robinson. A Course in the Theory of Groups. Springer New York, 2012. 
  23. [23] J.J. Rotman. An Introduction to the Theory of Groups. Springer, 1995. Zbl0810.20001
  24. [24] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329–334, 1990. 
  25. [25] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990. 
  26. [26] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990. 
  27. [27] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990. 
  28. [28] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5): 855–864, 1990. 
  29. [29] Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41–47, 1991. 
  30. [30] Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573–578, 1991. 
  31. [31] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990. 
  32. [32] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990. 
  33. [33] Katarzyna Zawadzka. Solvable groups. Formalized Mathematics, 5(1):145–147, 1996. 

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