Is the running maximum Markov?

Let $S_n$ be a symmetric random walk started at 0 with independent steps of $\pm 1$, each with probability $1/2$, and let $M_n = \max_{0 \le k \le n} S_k$ be its running maximum.

(a) Is $M_n$ a martingale?

(b) Is $M_n$ a Markov process?

(c) Is the pair $(S_n, M_n)$ a Markov process?