cp's OEIS Frontend

A058162 Number of labeled Abelian groups with a fixed identity.

%I A058162 #32 Mar 26 2022 22:05:09
%S A058162 1,1,1,4,6,60,120,1920,7560,90720,362880,13305600,39916800,1037836800,
%T A058162 10897286400,265686220800,1307674368000,66691392768000,
%U A058162 355687428096000,20274183401472000,202741834014720000
%N A058162 Number of labeled Abelian groups with a fixed identity.
%C A058162 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 A058162 That is, the number of Cayley tables. - _Artur Jasinski_, Mar 12 2008
%C A058162 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 A058162 Max Alekseyev, <a href="/A058162/b058162.txt">Table of n, a(n) for n = 1..100</a>
%H A058162 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 A058162 <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%F A058162 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 A058162 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 A058162 Cf. A000688, A058160, A058161, A058163.
%K A058162 nonn
%O A058162 1,4
%A A058162 _Christian G. Bower_, Nov 15 2000, Mar 12 2008
%E A058162 a(16) and a(21) corrected by _Max Alekseyev_, Sep 12 2019