Number of labeled Abelian groups of order n

I calculated the number of labeled Abelian groups of order $$N$$ (i.e., the number of distinct, abelian group laws on a set of $$N$$ elements). This sequence is given by OEIS A034382, but my solution differs at $$N=16$$.

Please point out mistakes or confirm my solution?

Let $$C_n$$ be a cyclic group of order $$n$$, $$Aut(G)$$ be an automorphism set of $$G$$.

Tthe number of labeled Abelian groups of order $$N$$ is $$\displaystyle{\sum \frac{N!}{\# Aut(G)}}$$ where G runs representative of isomorphic equivalence.

I got

$$\displaystyle{ \# Aut(C_{p^n}^k)=p^{(n-1)k^2}\prod_{j=0}^{k-1} (p^{k}-p^{j}) }$$

and

$$\displaystyle{ \# Aut(\prod_i C_{p^{n_i}}^{k_i}) =\prod_i \left( (p^{(n_i-1)k_i^2}\prod_{j=0}^{k_i-1} (p^{k_i}-p^{j})) ( \prod_{j\neq i} p^{\min(n_i,n_j)k_j} )^{k_i} \right) }$$.

From the fundamental theorem of finite abelian groups, There are 5 groups for $$N=16$$: $$C_{16}, C_2 \times C_8, C_4^2, C_2^2\times C_4, C_2^4$$.

Therefore, the number of groups that are isomorphic to each group is:

• $$C_{16}$$ ... $$\displaystyle{\frac{16!}{8}}$$
• $$C_2 \times C_8$$ ... $$\displaystyle{\frac{16!}{(1\times 2)\times (2\times 4)}}$$
• $$C_4^2$$ ... $$\displaystyle{\frac{16!}{16\times 3\times 2}}$$
• $$C_2^2\times C_4$$ ... $$\displaystyle{\frac{16!}{((3\times 2)\times 2^2)\times (2^2\times 2)}}$$
• $$C_2^4$$ ... $$\displaystyle{\frac{16!}{15\times 14\times 12\times 8}}$$

Sum of them is $$4250979532800$$. OEIS says $$4248755596800$$.

migrated from mathoverflow.netSep 13 at 12:07

This question came from our site for professional mathematicians.

• It's only off by one-twentieth of one percent – what's the big deal? – Gerry Myerson Sep 12 at 6:16
• I'm sorry that this should post to Mathematics stack exchange. I'll delete this question soon. – sugarknri Sep 12 at 10:59
• It may be an error in OEIS. In either case it's a good question, and I don't support the close votes. – Max Alekseyev Sep 12 at 12:38
• In case it's a good question, it would benefit from being better introduced... it starts with 3 sentences which look unrelated. The notion in the title is not defined. Also the title should not be considered as the 1st line of the post: the post should be meaningful without reading the title. – YCor Sep 12 at 23:17
• @YCor Thank you for the advice about the format. I change text. If the text is still not clear, It may be because of my English ability. – sugarknri Sep 13 at 3:35

There is a different formula for $$\#\mathrm{Aut}(\prod_i C_{p^{n_i}}^{k_i})$$ given in the paper Automorphisms of Finite Abelian Groups by Hillar and Rhea: $$\#\mathrm{Aut}(\prod_{t=1}^m C_{p^{e_t}}) = \prod_{t=1}^m (p^{d_t} - p^{t-1}) p^{e_t(m-d_t) + (e_t-1)(m-c_t+1)},$$ where $$1\leq e_1\leq e_2\leq \cdots\leq e_m$$, and $$c_t$$ and $$d_t$$ are the minimum and maximum of the set $$S_t := \{\ell\ :\ e_\ell=e_t\}$$, respectively.

Below I will show that OP's formula is equivalent to the Hillar-Rhea formula.

Let $$d_0:=0$$. It can be seen that the $$k_i$$'s are the nonzero elements of the multiset $$\{ d_1-d_0, d_2-d_1, \dots, d_m-d_{m-1}\}$$ and the $$n_i$$'s are the corresponding elements of $$\{e_1,e_2,\dots,e_m\}$$. Define $$s_0=0, s_1, \dots, s_q$$ be the indices such that $$k_i = d_{s_i} - d_{s_{i-1}}$$ and $$n_i = e_{s_i}$$. Vice versa, $$d_{s_i} = k_1+\dots+k_i$$ and $$c_{s_i} = d_{s_{i-1}}+1$$.

First consider these parts of the two formulas: $$\prod_{i=1}^q \prod_{j=0}^{k_i-1} (p^{k_i} - p^j) = \prod_{i=1}^q p^{k_i(k_i-1)/2} \prod_{j=0}^{k_i-1} (p^{k_i-j} - 1)$$ and $$\prod_{t=1}^m (p^{d_t} - p^{t-1}) = p^{m(m-1)/2}\prod_{t=1}^m (p^{d_t-t+1} - 1).$$ It is easy to see that the multisets $$\{ k_i - j : 0\leq j \leq k_i-1, 1\leq i\leq q \}$$ and $$\{ d_t - t +1\ :\ 1\leq t\leq m \}$$ are the same, since the $$t$$-th element in the sequence $$k_1 - 0, k_1 - 1, \dots, 1, k_2 - 0, k_2 - 1, \dots, 1, \dots$$ equals $$d_t-t+1$$.

Now it remains to prove the equality for the powers of $$p$$ in the two formulas, i.e. $$\sum_{i=1}^q \bigg(k_i(k_i-1)/2 + (n_i-1)k_i^2 + \sum_{j\ne i} \min(n_i,n_j)k_ik_j\bigg) = m(m-1)/2 + \sum_{t=1}^m \big(e_t(m-d_t) + (e_t-1)(m-c_t+1)\big).$$ In the l.h.s. we have $$\sum_{i=1}^q \sum_{j\ne i} \min(n_i,n_j)k_ik_j = 2\sum_{i=1}^q n_i k_i \sum_{j>i} k_j=2\sum_{i=1}^q n_i k_i (m-d_{s_i}).$$ In the r.h.s. we have $$\begin{split} \sum_{t=1}^m \big(e_t(m-d_t) + (e_t-1)(m-c_t+1)\big) &= \sum_{i=1}^q k_i\big(e_{s_i}(m-d_{s_i}) + (e_{s_i}-1)(m-d_{s_{i-1}})\big) \\ &= \sum_{i=1}^q k_i\big(n_i(m-d_{s_i}) + (n_i-1)(m-d_{s_{i-1}})\big)\\ &=\sum_{i=1}^q k_i\big(n_i(m-d_{s_i}) + (n_i-1)(m+k_i-d_{s_i})\big)\\ &=2\sum_{i=1}^q k_i n_i(m-d_{s_i}) + \sum_{i=1}^q \big( (n_i-1)k_i^2 - (m-d_{s_i})k_i\big). \end{split}$$ Finally, we notice that $$\sum_{i=1}^q k_i(k_i-1)/2 = m(m-1)/2 - \sum_{i=1}^q (m-d_{s_i})k_i$$ since $$m=k_1+k_2+\dots+k_q$$ and $$m-d_{s_i} = k_{i+1}+k_{i+1}+\dots+k_q$$. QED

So, we can conclude that OEIS A034382 did indeed contain an error in its 16-th term. Now it's corrected.