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].
@article{bwmeta1.element.doi-10_2478_forma-2013-0022, author = {Hiroshi Yamazaki and Hiroyuki Okazaki and Kazuhisa Nakasho and Yasunari Shidama}, title = {Isomorphisms of Direct Products of Cyclic Groups of Prime Power Order}, journal = {Formalized Mathematics}, volume = {21}, year = {2013}, pages = {207-211}, zbl = {1298.20068}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_forma-2013-0022} }
Hiroshi Yamazaki; Hiroyuki Okazaki; Kazuhisa Nakasho; Yasunari Shidama. Isomorphisms of Direct Products of Cyclic Groups of Prime Power Order. Formalized Mathematics, Tome 21 (2013) pp. 207-211. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_forma-2013-0022/
[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] | Zbl 1277.20067
[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. | Zbl 06213858
[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. | Zbl 1298.55008
[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] | Zbl 1277.20068
[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.