Abstract:The main results concern commutativity of Hewitt-Nachbin realcompactification or Dieudonn\'e completion with products of topological groups. It is shown that for every topological group $G$ that is not Dieudonn\'e complete one can find a Dieudonn\'e complete group $H$ such that the Dieudonn\'e completion of $G\times H$ is not a topological group containing $G\times H$ as a subgroup. Using Korovin's construction of $G_\delta $-dense orbits, we present some examples showing that some results on topological groups are not valid for semitopological groups.
Keywords: topological group, Dieudonn\'e completion, PT-group, realcompactness, \newline Moscow space, $C$-embedding, product
AMS Subject Classification: 22A05, 54H11, 54D35, 54D60