Isomorphisms of Direct Products of Cyclic Groups of Prime Power Order

Hiroshi Yamazaki; Hiroyuki Okazaki; Kazuhisa Nakasho; Yasunari Shidama

Formalized Mathematics (2013)

  • Volume: 21, Issue: 3, page 207-211
  • ISSN: 1426-2630

Abstract

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In this paper we formalized some theorems concerning the cyclic groups of prime power order. We formalize that every commutative cyclic group of prime power order is isomorphic to a direct product of family of cyclic groups [1], [18].

How to cite

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Hiroshi Yamazaki, et al. "Isomorphisms of Direct Products of Cyclic Groups of Prime Power Order." Formalized Mathematics 21.3 (2013): 207-211. <http://eudml.org/doc/267126>.

@article{HiroshiYamazaki2013,
abstract = {In this paper we formalized some theorems concerning the cyclic groups of prime power order. We formalize that every commutative cyclic group of prime power order is isomorphic to a direct product of family of cyclic groups [1], [18].},
author = {Hiroshi Yamazaki, Hiroyuki Okazaki, Kazuhisa Nakasho, Yasunari Shidama},
journal = {Formalized Mathematics},
keywords = {formalization of the commutative cyclic group; prime power set; formalized mathematics; formalization of groups; products of finite cyclic groups; finite Abelian groups; direct products},
language = {eng},
number = {3},
pages = {207-211},
title = {Isomorphisms of Direct Products of Cyclic Groups of Prime Power Order},
url = {http://eudml.org/doc/267126},
volume = {21},
year = {2013},
}

TY - JOUR
AU - Hiroshi Yamazaki
AU - Hiroyuki Okazaki
AU - Kazuhisa Nakasho
AU - Yasunari Shidama
TI - Isomorphisms of Direct Products of Cyclic Groups of Prime Power Order
JO - Formalized Mathematics
PY - 2013
VL - 21
IS - 3
SP - 207
EP - 211
AB - In this paper we formalized some theorems concerning the cyclic groups of prime power order. We formalize that every commutative cyclic group of prime power order is isomorphic to a direct product of family of cyclic groups [1], [18].
LA - eng
KW - formalization of the commutative cyclic group; prime power set; formalized mathematics; formalization of groups; products of finite cyclic groups; finite Abelian groups; direct products
UR - http://eudml.org/doc/267126
ER -

References

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  1. [1] Kenichi Arai, Hiroyuki Okazaki, and Yasunari Shidama. Isomorphisms of direct products of finite cyclic groups. Formalized Mathematics, 20(4):343-347, 2012. doi:10.2478/v10037-012-0038-5.[Crossref] Zbl1277.20067
  2. [2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 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] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990. 
  8. [8] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990. 
  9. [9] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990. 
  10. [10] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990. 
  11. [11] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990. 
  12. [12] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990. 
  13. [13] Artur Korniłowicz. The product of the families of the groups. Formalized Mathematics, 7(1):127-134, 1998. Zbl1298.55008
  14. [14] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990. 
  15. [15] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990. 
  16. [16] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990. 
  17. [17] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1): 103-108, 1993. 
  18. [18] 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.[Crossref] Zbl1277.20068
  19. [19] Dariusz Surowik. Isomorphisms of cyclic groups. Some properties of cyclic groups. Formalized Mathematics, 3(1):29-32, 1992. 
  20. [20] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990. 
  21. [21] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003. 
  22. [22] Andrzej Trybulec. Many sorted sets. Formalized Mathematics, 4(1):15-22, 1993. 
  23. [23] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990. 
  24. [24] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990. 
  25. [25] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5): 855-864, 1990. 
  26. [26] Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955-962, 1990. 
  27. [27] Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41-47, 1991. 
  28. [28] Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573-578, 1991. 
  29. [29] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990. 
  30. [30] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990. 
  31. [31] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990. 

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