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21.10 (2026)

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We call a group presentation finite if it represents a finite group. We say that a presentation is just finite if it is finite and is no longer finite on removal of any relation from it. Is it true that every finite group has a just finite presentation?

Note that if a group has a balanced presentation, then it is just finite. A similar argument can be applied for some $p$-groups using the Golod–Shafarevich inequality.

Contributor: Y. Barnea

Progress

Yes, it is true (M. Lackenby, Preprint, 2026, https://arxiv.org/abs/2605.10402).

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