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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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Online
ISSN
1898-9934
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Volume 21, Issue 3 (Oct 2013)

Issues

Isomorphisms of Direct Products of Cyclic Groups of Prime Power Order

Hiroshi Yamazaki
  • Shinshu University Nagano, Japan
/ Hiroyuki Okazaki
  • Shinshu University Nagano, Japan
/ Kazuhisa Nakasho
  • Shinshu University Nagano, Japan
/ Yasunari Shidama
  • Shinshu University Nagano, Japan

Summary

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].

Keywords: formalization of the commutative cyclic group; prime power set

  • [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]

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

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

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

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

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

  • [7] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.

  • [8] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.

  • [9] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.

  • [10] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.

  • [11] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.

  • [12] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.

  • [13] Artur Korniłowicz. The product of the families of the groups. Formalized Mathematics, 7(1):127-134, 1998.

  • [14] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.

  • [15] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.

  • [16] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.

  • [17] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1): 103-108, 1993.

  • [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]

  • [19] Dariusz Surowik. Isomorphisms of cyclic groups. Some properties of cyclic groups. Formalized Mathematics, 3(1):29-32, 1992.

  • [20] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990.

  • [21] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.

  • [22] Andrzej Trybulec. Many sorted sets. Formalized Mathematics, 4(1):15-22, 1993.

  • [23] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.

  • [24] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.

  • [25] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5): 855-864, 1990.

  • [26] Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955-962, 1990.

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

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

  • [29] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

  • [30] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.

  • [31] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

About the article

Published in Print: 2013-10-01


Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.2478/forma-2013-0022. Export Citation

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