Isomorphisms of Direct Products of Finite Cyclic Groups

Open access

Summary

In this article, we formalize that every finite cyclic group is isomorphic to a direct product of finite cyclic groups which orders are relative prime. This theorem is closely related to the Chinese Remainder theorem ([18]) and is a useful lemma to prove the basis theorem for finite abelian groups and the fundamental theorem of finite abelian groups. Moreover, we formalize some facts about the product of a finite sequence of abelian groups.

If the inline PDF is not rendering correctly, you can download the PDF file here.

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

  • [2] Grzegorz Bancerek. K¨onig’s theorem. Formalized Mathematics 1(3):589-593 1990.

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

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

  • [5] Czesław Bylinski. Binary operations. Formalized Mathematics 1(1):175-180 1990.

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

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

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

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

  • [10] Czesław Bylinski. The sum and product of finite sequences of real numbers. FormalizedMathematics 1(4):661-668 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. On the real valued functions. Formalized Mathematics 13(1):181-187 2005.

  • [14] Eugeniusz Kusak Wojciech Leonczuk and Michał Muzalewski. Abelian groups fields and vector spaces. Formalized Mathematics 1(2):335-342 1990.

  • [15] Anna Lango and Grzegorz Bancerek. Product of families of groups and vector spaces. Formalized Mathematics 3(2):235-240 1992.

  • [16] Hiroyuki Okazaki Noboru Endou and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics 19(1):51-59 2011 doi: 10.2478/v10037-011-0009-2.

  • [17] Christoph Schwarzweller. The ring of integers Euclidean rings and modulo integers. Formalized Mathematics 8(1):29-34 1999.

  • [18] Christoph Schwarzweller. Modular integer arithmetic. Formalized Mathematics 16(3):247-252 2008 doi:10.2478/v10037-008-0029-8.

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

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

  • [21] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics 1(2):291-296 1990.

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

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

Search
Journal information
Impact Factor


CiteScore 2018: 0.42

SCImago Journal Rank (SJR) 2018: 0.111
Source Normalized Impact per Paper (SNIP) 2018: 0.169

Target audience:

researchers in the fields of formal methods and computer-checked mathematics

Metrics
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 344 132 15
PDF Downloads 131 55 7