Relation gap
Groups, moves and rules · relation gap
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The situation where the least number of relations of a group is larger than the least number of generators of its relation module. Karl Gruenberg raised the question whether this happens in 1976, and in 1979 it entered Wall’s list of problems (problem B15). In the game “step and lift” the player needs two rules while the accountant needs one entry: the paper arXiv 2610.23637 of 16 October 2026 settled the question.
Two ways to count rules
- Player · A rule is a route on the map, the order of moves matters. The least number of relations; in “step and lift” it is 2.
- Accountant · A rule is a ledger entry, the order of moves is not remembered. The least number of generators of the relation module; in the same game it is 1.
Source: arXiv 2610.23637: Two-generator presentations with a relation gap
Read more: How many rules does a game need: the relation gap
Where it appears · 1
- How many rules does a game need · Research