There are No Product and Subgroup Theorems for the Covering Dimension of Topological Groups

Strongly zero-dimensional topological groups $G_1$, $G_2$, and $G$ such that $G_1\times G_2$ has positive covering dimension and $G$ contains a closed subgroup of positive covering dimension are constructed. Moreover, all finite powers of $G_1$ are Lindelöf and $G_2$ is second-countable. An example of a strongly zero-dimensional space $X$ whose free, free Abelian, and free Boolean topological groups have positive covering dimension is also given.