Let $G$ be a finite abelian group. Write $G\approx \mathbb{Z}/p\_1^{i\_1}\mathbb{Z}\times\dots \mathbb{Z}/p\_m^{i\_m}\mathbb{Z}$, with $m\ge 0$, $p\_1,\dots,p\_m$ primes (not necessarily distinct) and $i\_k\ge 1$ for all $k$. 


For $n\ge 0$, is it true that there is an injective homomorphism $G\rightarrow S\_n$ if and only if $p\_1^{i\_1}+\dots+p\_m^{i\_m}\leq n$?



A criterion for finite abelian group to embed into a symmetric group
Yes, this is true (provided we add the hypothesis $i\_1, \ldots, i\_n \ge 1$), and not hard to prove by considering centralizers of products of cycles in symmetric groups.


According to <http://www-history.mcs.st-andrews.ac.uk/Biographies/Povzner.html>, the minimum degree of a permutation representation of an arbitrary abelian group was found in A. Ya. Povzner, *Finding groups of permutations of least degree which are isomorphic to a
given Abelian group*, Kharkov Mat. Obshch., **14** (1937) 151–158. (I have not been able to obtain this paper to check.)


Theorem 4.1 in Kay Jin Lim, *Specht modules with abelian vertices*, 
J. Algebraic Combin. **35** (2012) 157–171 states your conjectured result in the special case when all the primes are the same with a reference to Povzner's paper.


