Homogeneous Markov chains which are defined on a homogeneous measurable space $(X,G,\mathfrak{B})$ are considered. Here X is a set of states of the chain, G a transitive group of transformations of the space X, $\mathfrak{B}$ the $\sigma $-algebra of subsets of X. If the transition function $P(x,E)$ has the property $P(x,E)P(gx,gE)$ for arbitrary $x \in X,g \in G,E \in \mathfrak{B}$, the Markov chain is called invariant on the space $(X,G,\mathfrak{B})$. We say that the Markov chain is strictly regular on the set $H,H \subseteq X$, if 1) $H = \bigcup\limits_{\alpha \in S} {E_\alpha } $ (S is a system of values of the index $\alpha $), \[ E_\alpha \cap E_\beta = 0{\text{ at }}\alpha \ne \beta ,\quad \alpha ,\beta \in S,\] 2) for an arbitrary index $\alpha \in S$\[ E_\alpha = \bigcup\limits_{i = 1}^{d\alpha } {E_{\alpha ,i} } ,\quad E_{\alpha ,i} \cap E_{\alpha ,j} = 0,\quad i \ne j;i,j \leqq dx, \] and for all $x \in E_{\alpha ,i} $, $E \in \mathfrak{B}$ there exists a limit: \[ \mathop {\lim }\limits_{n \to \infty } P^{(nd_\alpha + j - i)} (x,E) = \varphi _{\alpha ,j} (E) \] which is a probability measure in $E \in \mathfrak{B}$. A criterion is given in the paper for determining whether or not a Markov chain invariant on the space $(X,G,\mathfrak{B})$ is strictly regular; some other asymptotic properties of $P^{(n)} (x,E)$ are estab lished.