# \$ n \$ product groups-periodicals

We call a group (infinite) $$n$$periodic product 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 $$prod_ {i = 1} ^ k G not free G$$.

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