The restricted wreath product of groups and iswhere acts on the finitely supported functions by translation.
The lamplighter group is the restricted wreath product . It is generated by one lamp switch and the translation of the integer line.
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In group theory, the wreath product is a specific way to construct a new group from two given groups. It is particularly useful in the study of permutation groups and can be thought of as a form of "combining" groups while retaining certain properties.