Motivation. Let us call a group $G = (G,\cdot)$ (product-)reducible if there are groups $H_1, H_2$, each having more than $1$ element, with $G \cong H_1\times H_2$. Otherwise, $G$ is said to be irreducible. If $G$ is finite, then $G$ is isomorphic to a product of irreducible groups. This is because in the finite case, proper subgroups of a group have strictly smaller cardinality, so you can't have infinitely descending chains of proper subgroups. - In the infinite case, things may look different.

Infinite case. Is there an infinite group $G$ such that whenever $\kappa \geq 0$ is a cardinal and $G_\alpha$ are non-trivial groups for $\alpha\in\kappa$ such that $G \cong \prod_{\alpha\in\kappa}G_\alpha$, then for all $\alpha\in\kappa$, the group $G_\alpha$ is reducible?