Isomorphisms of Direct Products of Cyclic Groups of Prime Power Order

Hiroshi Yamazaki; Hiroyuki Okazaki; Kazuhisa Nakasho; Yasunari Shidama

doi:10.2478/forma-2013-0022

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Isomorphisms of Direct Products of Cyclic Groups of Prime Power Order

Formalized Mathematics

Volume 21 (2013): Issue 3 (October 2013)

By: Hiroshi Yamazaki, Hiroyuki Okazaki, Kazuhisa Nakasho and Yasunari Shidama

Open Access

|Oct 2013

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].

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. Monoids. Formalized Mathematics, 3(2):213-225, 1992.
Search in Google Scholar
[4] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
Search in Google Scholar
[5] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
Search in Google Scholar
[6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
Search in Google Scholar
[7] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
Search in Google Scholar
[8] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
Search in Google Scholar
[9] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
Search in Google Scholar
[10] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
Search in Google Scholar
[11] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
Search in Google Scholar
[12] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.
Search in Google Scholar
[13] Artur Korniłowicz. The product of the families of the groups. Formalized Mathematics, 7(1):127-134, 1998.
Search in Google Scholar
[14] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.
Search in Google Scholar
[15] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.
Search in Google Scholar
[16] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.
Search in Google Scholar
[17] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1): 103-108, 1993.
Search in Google Scholar
[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-2013-0007.10.2478/forma-2013-0007
Search in Google Scholar
[19] Dariusz Surowik. Isomorphisms of cyclic groups. Some properties of cyclic groups. Formalized Mathematics, 3(1):29-32, 1992.
Search in Google Scholar
[20] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990.
Search in Google Scholar
[21] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.
Search in Google Scholar
[22] Andrzej Trybulec. Many sorted sets. Formalized Mathematics, 4(1):15-22, 1993.
Search in Google Scholar
[23] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
Search in Google Scholar
[24] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.
Search in Google Scholar
[25] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5): 855-864, 1990.
Search in Google Scholar
[26] Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955-962, 1990.
Search in Google Scholar
[27] Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41-47, 1991.
Search in Google Scholar
[28] 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
[29] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
Search in Google Scholar
[30] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
Search in Google Scholar
[31] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.2478/forma-2013-0022 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630

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Language: English

Page range: 207 - 211

Published on: Oct 1, 2013

Published by: University of Białystok

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

formalization of the commutative cyclic group,

prime power set

Related subjects:

Mathematics,

Logic and set theory,

Computer sciences,

Artificial intelligence

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

Volume 21 (2013): Issue 3 (October 2013)