Let $\Omega$ be a finite state space, and let $(X_n)_{n\geq1}$ be a Markov chain on $\Omega$. Suppose $(X_n)$ is irreducible, i.e. $\forall \omega,\omega'\in\Omega$ $\exists N\geq 1$ $P(X_N=\omega'|X_1=\omega)>0$.
I would like to prove that $(X_n)$ is aperiodic if and only if: $\exists N\geq 1$ $\forall \omega,\omega'\in\Omega$ $P(X_N=\omega'|X_1=\omega)>0$. $(P)$
[$(P)\Rightarrow$ aperiodic] seems clear, indeed let $\omega\in \Omega$ : $(P)$ implies that there exists $N\geq1$ such that $P(X_N=\omega|X_1=\omega)>0$, but also $P(X_{N+1}=\omega|X_1=\omega)>0$. We conclude since $PGCD(N,N+1)=1$.
I cannot manage to prove the converse.