Abstract:We will show that under ${MA}_{countable}$ for each $k \in \Bbb N$ there exists a group whose $k$-th power is countably compact but whose $2^k$-th power is not countably compact. In particular, for each $k \in \Bbb N$ there exists $l \in [k,2^k)$ and a group whose $l$-th power is countably compact but the $l+1$-st power is not countably compact.
Keywords: countable compactness, ${MA}_{countable}$, topological groups, finite powers
AMS Subject Classification: 54D20, 54H11, 54B10, 54A35, 22A05