## $n$-product-periodic topological spaces

We call an topological space $$(X,\tau)$$ $$n$$product-periodic for an integer $$n\geq 3$$ if $$\prod_{i=1}^n X \cong X$$ but for all integers $$k$$ with $$2\leq k\leq n-1$$ we have $$\prod_{i=1}^k X \not\cong X$$.

Is there an integer $$n\geq 3$$ such that there is an $$n$$-product-periodic space, and is there an integer $$m\geq 3$$ such that there is no $$m$$-product-periodic space?

## $n$-product-periodic groups

We call an (infinite) group $$n$$product-periodic for an integer $$n\geq 3$$ if $$\prod_{i=1}^n G \cong G$$ but for all integers $$k$$ with $$2\leq k\leq n-1$$ we have $$\prod_{i=1}^k G \not\cong G$$.

Is there an integer $$n\geq 3$$ such that there is a group $$G$$ such that $$G$$ is $$n$$-product-periodic, and is there an integer $$m\geq 3$$ such that there is no group $$G$$ such that $$G$$ is $$m$$-product-periodic?