How many different nonisomorphic Abelian groups of order n are possible ??

closed as too localized by Alex Bartel, Qiaochu Yuan, Pete L. Clark, David Hansen, HJRW Feb 10 '11 at 17:22
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Depending on the interpretation of the question, I either somewhat agree or disagree with the commenter. In view of this, I would like to encourage the OP to (in the future) be more detailed when asking questions. The question of determining the number of nonisomorphic finite abelian groups of order $n$ for a fixed $n$ is, assuming one is able to determine the factorization of $n$ into primes, not too complicated a task if one is aware of the structure theorem of finite abelian groups (STFA). The problem is via the STFA actually the problem of determining in how many ways one can factor $n$ as$n_1 \dots n_r$ with $1<n_1 \mid \dots n_r$ ; or possibly simpler first decomposing into prime powers / $p$groups and only then carrying out this consideration. In any case, this translates to the problem in how many ways one can partition the multiplicties of the primes occuring in $n$. Or, explicitly, the number for $n$ is $\prod_{i=1}^k P(e_i)$ where $n=p_1^{e_1}\dots p_n^{e_k}$ is the factorization into primepowers and $P$ denotes the number of partitions. In particular, on the one hand there cannot be an absolute bound (independent of $n$) on the number of isomorphy classes of finite ablian groups of a given order; on the other hand, there exists arbitrarily large $n$ where there is a unique equivalence class, namely precisely if $n$ is squarefree. However, there is also the question of counting all isomorphy classes of finite abelian groups (up to a given order). Let $A(x)$ denote the number of isomorphy classes of finite abelian groups of order at most $x$. The question of determining the asymptotic of $A(x)$ was first considered by Erd{\H o}s and Szekeres. It is (at least) known that $A(x) = C_1 x + C_2 x^{1/2}+ C_3 x^{1/3} + \Delta(x)$ where $\Delta(x)= O(x^{55/219} (\log x)^7)$ and $C_i = \prod_{j=1, j\neq i}^{\infty} \zeta(j/i)$ and $\zeta$ is Riemann's zetafunction. The 'at least' means that this is the best result I know (due to Sargos and Wu, 2000), but in view of the fact that the problem of improving estimates for $\Delta(x)$ was considered a lot it might well be the case that since 2000 there were further improvements I am unaware of. 

