A graph puzzle Puz(G) of a graph G is defined as follows. A configuration of Puz(G) is a bijection from the set of vertices of a board graph to the set of vertices of a pebble graph, both graphs being isomorphic to some input graph G. A move of pebbles is defined as exchanging two pebbles which are adjacent on both a board graph and a pebble graph. For a pair of configurations f and g, we say that f is equivalent to g if f can be transformed into g by a finite sequence of moves.Let Aut(G) be the automorphism group of G, and let 1(G) be the unit element of Aut(G). The pebble exchange group of G, denoted by Peb(G), is defined as the set of all automorphisms f of G such that 1(G) and f are equivalent to each other.In this paper, some basic properties of Peb(G) are studied. Among other results, it is shown that for any connected graph G, all automorphisms of G are contained in Peb(G(2)), where G(2) is a square graph of G. (C) 2021 Elsevier Ltd. All rights reserved.