Centralizer
Groups, moves and rules · all “friends” of a move
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All moves that can be swapped with a given move: its “friends”. In the game “step and lift” the friends of the half-circle x³ include two steps in a row x² and the hook t·x²·t⁻¹, but they are not friends with each other. Newman’s theorem (1973): in a group with one relation and a move that returns to the start after several repeats, the friends of any move form a chain of repeats of a single move and therefore are friends with each other. This is why this game has more than one rule.
Source: Wikipedia: Centralizer and normalizer
Read more: Why one rule is not enough for the player: friends are not friends
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- How many rules does a game need · Research