A058162 - OEIS

%I #32 Mar 26 2022 22:05:09

%S 1,1,1,4,6,60,120,1920,7560,90720,362880,13305600,39916800,1037836800,

%T 10897286400,265686220800,1307674368000,66691392768000,

%U 355687428096000,20274183401472000,202741834014720000

%N Number of labeled Abelian groups with a fixed identity.

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

%C That is, the number of Cayley tables. - _Artur Jasinski_, Mar 12 2008

%C 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.

%H Max Alekseyev, <a href="/A058162/b058162.txt">Table of n, a(n) for n = 1..100</a>

%H C. J. Hillar, D. Rhea. <a href="https://www.jstor.org/stable/27642365">Automorphisms of finite Abelian groups</a>. American Mathematical Monthly 114:10 (2007), 917-923. Preprint <a href="https://arxiv.org/abs/math/0605185">arXiv:math/0605185</a> [math.GR]

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%F 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).

%e 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.

%Y Cf. A000688, A058160, A058161, A058163.

%K nonn

%O 1,4

%A _Christian G. Bower_, Nov 15 2000, Mar 12 2008

%E a(16) and a(21) corrected by _Max Alekseyev_, Sep 12 2019