A finite group $G$ and fixed $k\geq 1$ where for every $n\geq 1$, the $n$-direct product $G^n=G\times\dots\times G$ is $k$-generated?

Question

Does exist a finite group $G$ and fixed $k \geq 1$ such that the $n$-direct product $G^n = G \times \dots \times G$ is $k$-generated for every $n \geq 1$?

I suspect the answer is no. Does exist a simple proof of this fact (if true)?

Answer 1

No. For given $k$ and sufficiently large $n$, the projections of all $k$ putative generators would be equal on two different components, and so they could not generate the full direct product.

To be more precise, if $h(k)$ is the number of surjective homomorphisms from the free group $F_k$ of rank $k$ to $G$ up to equivalence under automorphisms of $G$, then $G^{h(k)}$ is $k$-generated, but $G^{h(k)+1}$ is not. A well-known example (due I think to Philip Hall) is $A_5^{19}$ is $2$-generated but $A_5^{20}$ is not.