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In this paper we search for conditions on a countably compact(pseudo-compact) topological semigroup under which: (i) each maximal subgroup$H(e)$ in $S$ is a (closed) topological subgroup in $S$; (ii) the Clifford part$H(S)$(i.e. the union of all maximal subgroups) of the semigroup $S$ is aclosed subset in $S$; (iii) the inversion $\operatorname{inv}\colon H(S)\toH(S)$ is continuous; and (iv) the projection $\pi\colon H(S)\to E(S)$,$\pi\colon x\longmapsto xx^{-1}$, onto the subset of idempotents $E(S)$ of $S$,is continuous.

The continuity of the inversion and the structure of maximal subgroups in countably compact topological semigroups