Let $S_{n}:= S_0 + \sum_{i=1}^{n}X_i$ be a simple random walk, $X_i$ are independent random variables with $P[X_i=1] = p, P[X_i = -1] = 1-p$.
Let $M_n:=\max\{S_0, \dots, S_n\}$.
The task at hand is to prove or disprove that the following items are Markov-chains.
(a) $(M_n)_{n\in \mathbb{N}_0}$
(b) $(|S_n|)_{n\in \mathbb{N}_0}$
(c) $(S_n, M_n)_{n\in \mathbb{N}_0}$
I was able to show that (a) is no Markov-chain with a counter example but I don't know how to show the other two.
(c) is a Markov chain, since the transition at step $n+1$ only depends on the value of the sum up to $n$ (the endpoint of the random walk) and the maximum thus far.
(b) is not a markov chain unless $p=\frac{1}{2}$ and a counter-example is to take $n=1$; then $|S_n| = 1$ but $P(|S_2|=2) = p$ if the first step was to $-1$ but is $1-p$ if the first step was to $+1$.