One of the central research tasks in the theory of compact semitopological semigroups is to explore the structure of semitopological compactifications of groups. The present paper contributes the following structural information about such compactifications $S$: (i) the semigroup $S$ is (algebraically) a semilattice of groups if (and only if) it is a union of groups; (ii) if $S$ is a regular semigroup then the idempotents form a topological semilattice and the action of this semilattice on $S$ by multiplication is jointly continuous; (iii) if the set of all units (= invertible elements in $S$) which lie in a compact subgroup of $S$ is dense in the group of units then $sS=Ss$ for every $s\in S$ and, in particular, all idempotents of $S$ are central. Results (i) and (iii) solve two problems posed by the reviewer [in Recent developments in the algebraic, analytical, and topological theory of semigroups (Oberwolfach, 1981), 215--238, Lecture Notes in Math., 998, Springer, Berlin, 1983; MR0724631 (84m:22004)]. (Wolfgang A. F. Ruppert)