Let $X$ be an irreducible homogeneous Markov chain $(\mu,P)$ with state space $S$. The following statements should be proved or disproved:
a) If a $x\in S$ with $P(x,x)>0$ exists, then $X$ is aperiodic.
b) If $X$ is aperiodic, then there exists a $x\in S$ with $P(x,x) >0$.
c) If $S$ is finite and $X$ is periodic with period $2$, then $-1$ is an eigenvalue of $P$.
d) If $S$ is finite and $-1$ is an eigenvalue of $P$, then $X$ is periodic with period $2$.
I think the first three statements are true and the last one is false, but I have no idea how to prove or disprove them.