Isomorphisms of Direct Products of Finite Commutative Groups

Hiroyuki Okazaki; Hiroshi Yamazaki; Yasunari Shidama

doi:10.2478/forma-2013-0007

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Isomorphisms of Direct Products of Finite Commutative Groups

Formalized Mathematics

Volume 21 (2013): Issue 1 (January 2013)

By: Hiroyuki Okazaki, Hiroshi Yamazaki and Yasunari Shidama

Open Access

|Aug 2013

Abstract

We have been working on the formalization of groups. In [1], we encoded some theorems concerning the product of cyclic groups. In this article, we present the generalized formalization of [1]. First, we show that every finite commutative group which order is composite number is isomorphic to a direct product of finite commutative groups which orders are relatively prime. Next, we describe finite direct products of finite commutative groups

References

[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/v10037-012-0038-5.10.2478/v10037-012-0038-5
Search in Google Scholar
[2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
Search in Google Scholar
[3] Grzegorz Bancerek. K¨onig’s theorem. Formalized Mathematics, 1(3):589-593, 1990.
Search in Google Scholar
[4] Grzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213-225, 1992.
Search in Google Scholar
[5] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
Search in Google Scholar
[6] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
Search in Google Scholar
[7] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
Search in Google Scholar
[8] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
Search in Google Scholar
[9] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
Search in Google Scholar
[10] Czesław Bylinski. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990.
Search in Google Scholar
[11] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
Search in Google Scholar
[12] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
Search in Google Scholar
[13] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
Search in Google Scholar
[14] Artur Korniłowicz. The product of the families of the groups. Formalized Mathematics, 7(1):127-134, 1998.
Search in Google Scholar
[15] Artur Korniłowicz and Piotr Rudnicki. Fundamental Theorem of Arithmetic. FormalizedMathematics, 12(2):179-186, 2004.
Search in Google Scholar
[16] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.
Search in Google Scholar
[17] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.
Search in Google Scholar
[18] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1): 103-108, 1993.
Search in Google Scholar
[19] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990.
Search in Google Scholar
[20] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.
Search in Google Scholar
[21] Andrzej Trybulec. Many sorted sets. Formalized Mathematics, 4(1):15-22, 1993.
Search in Google Scholar
[22] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
Search in Google Scholar
[23] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.
Search in Google Scholar
[24] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5): 855-864, 1990.
Search in Google Scholar
[25] Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955-962, 1990.
Search in Google Scholar
[26] Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. FormalizedMathematics, 2(1):41-47, 1991.
Search in Google Scholar
[27] Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573-578, 1991.
Search in Google Scholar
[28] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
Search in Google Scholar
[29] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.2478/forma-2013-0007 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630

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Language: English

Page range: 65 - 74

Published on: Aug 23, 2013

Published by: University of Białystok

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Related subjects:

Mathematics,

Logic and set theory,

Computer sciences,

Artificial intelligence

© 2013 Hiroyuki Okazaki, Hiroshi Yamazaki, Yasunari Shidama, published by University of Białystok
This work is licensed under the Creative Commons License.

Volume 21 (2013): Issue 1 (January 2013)