Self-similar group action

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Suppose G is a group and X is a finite set which we think of as an alphabet. Let X^* be the set of all words on X (viewed as a monoid), and let \cdot denote a faithful group action of G on X^* as a set (in particular, the action is not an action by monoidal automorphisms). We say that \cdot is a self-similar group action if for every g \in G and x \in X, there exist h \in G and y \in X such that:

\! g \cdot (xw) = y (h\cdot w)

The multiplication by concatenation is happening in the monoid X^*, where it is literally symbol concatenation.

The term self-similar group is typically used for a group that admits a self-similar group action, but in its usual usage, the action is understood to be implicitly specified, even though it is not part of the abstract group structure.