A058162 Number of labeled Abelian groups with a fixed identity.

1, 1, 1, 4, 6, 60, 120, 1920, 7560, 90720, 362880, 13305600, 39916800, 1037836800, 10897286400, 265686220800, 1307674368000, 66691392768000, 355687428096000, 20274183401472000, 202741834014720000

Offset: 1

Author

Christian G. Bower, Nov 15 2000, Mar 12 2008

Keywords

• nonn

Comments

The distinction here between labeled and unlabeled Abelian groups is analogous to the distinction between unlabeled rooted trees (A000081) and labeled rooted trees (A000169).

That is, the number of Cayley tables. - Artur Jasinski, Mar 12 2008

Number of Latin squares in dimension n with first row and first column 1,2,3 ..., n which are associative and commutative (Abelian). Each of these squares is isomorphic with the Cayley table of one of the existed Abelian group in dimension n. - Artur Jasinski, Nov 02 2005. Cf. A111341.

Examples

			The 2 unlabeled Abelian groups of order 4 are C4 and C2^2. The 4 labeled Abelian groups whose identity is "0" consist of 3 of type C4 (where the nongenerator can be "2", "3", or "4") and 1 of type C2^2.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..100
• C. J. Hillar, D. Rhea. Automorphisms of finite Abelian groups. American Mathematical Monthly 114:10 (2007), 917-923. Preprint arXiv:math/0605185 [math.GR]
• Index entries for sequences related to groups

Crossrefs

Cf. A000688, A058160, A058161, A058163.

Formula

a(n) = A034382(n) / n. Formula for A034382 is based on the fundamental theorem of finite Abelian groups and the formula given by Hillar and Rhea (2007).

Extensions

a(16) and a(21) corrected by Max Alekseyev, Sep 12 2019