Some old and basic facts about random walks on groups

This note contains old instead of new results about random walks on groups, which may serve as a small supplement to the author’s monograph Random Walks on Infinite Graphs and Groups'' (Cambridge Univ. Press 2000/2009). First, we exhibit a basic exercise on the periodicity classes of random walk. The second topic concerns some basics on ratio limits for random walks, which had been published only’’ in German in the 1970ies.