This research demonstrates automaton groups for finite groups, indicating a link to lamplighter groups.
For any nontrivial finite group X we construct a reversible automaton whose state set and alphabet coincide with X, with transition and output functions defined by the left regular action of X. We then study the associated automaton group GX and prove, in particular, that GX contains the lamplighter group A Z, where A denotes the center of X.
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A.S. Oliynyk (2025) studied this question.
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