Least symmetric group having a certain Abelian group as subgroup

Question

Given an Abelian group $G\simeq\bigoplus_{k}\mathbb Z_{p^{n_k}_{k}}$, where $p_1\leq p_2\leq ...$ are primes, how to calculate the least symmetric group $S_n$ having a subgroup isomorphic to $G$?

Answer 1

The answer is $n = \sum_{k} p_k^{n_k}$.

That follows from the more general result that, for finite nilpotent groups $G$ and $H$, ${\rm mindeg}(G \times H) = {\rm mindeg}(G) + {\rm mindeg}(H)$. I haven't checked the reference myself, but it is proved in D. Wright. Degrees of minimal embeddings of some direct products. Amer. J. Math., 97:897–903, 1975.

See also Neil Saunders, STRICT INEQUALITIES FOR MINIMAL DEGREES OF DIRECT PRODUCTS, Bull. Aust. Math. Soc. 79 (2009), 23–30 for examples that show that this equality is not true for all finite groups. One example is the transitive group $C_2 \wr C_5$ of degree $10$, which is the direct product of its centre of order $2$ and a subgroup of index $2$, which also have minimal degree $10$.