Mathematicians are trying to classify every possible way a finite group can act on a set of objects without repeating itself—a "binary action"—and they are now within reach of a complete list. This matters because finite groups are the mathematical language of symmetry, appearing everywhere from the structure of crystals to the error-correction codes that keep data intact. For 25 years, researchers have known that binary actions form the most basic building blocks of how groups behave, but no one has been able to catalogue them all. A major breakthrough came when the team proved a 20-year-old conjecture about the simplest class of these actions, the primitive ones. Now they aim to finish the job for all finite groups. If successful, the classification will unlock a powerful model-theoretic framework developed by Gregory Cherlin, allowing mathematicians to understand finite permutation groups in a unified way. This is fundamental science with no immediate practical application. But similar classifications—like the complete list of finite simple groups—have underpinned advances in cryptography, quantum computing, and network theory for decades. A deeper understanding of symmetry’s most basic actions could eventually reshape how we design everything from secure communications to molecular simulations.
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This research project is in the field of group theory which is the mathematical study of symmetry. The purpose of this research project is to classify the binary actions of finite groups. A faithful group action is binary if and only if it is isomorphic to the action on vertices of the automorphism group of a homogeneous edge-coloured graph. The notion of a binary action was introduced by the model theorist, Gregory Cherlin, who described an "organising principle" for the universe of finite permutation groups, via actions on homogeneous relational structures. From this point of view, binary actions form the most basic examples of such actions. Still, basic or not, 25 years later we still do not know the binary actions of the finite groups. A major breakthrough in this regard was achieved when 3 members of the proposed research team were able to complete a classification of the finite primitive binary permutation groups. Primitive groups are a special class of permutation groups and Cherlin himself gave a conjecture as to what the binary ones should look like. It took 20 years, but we now know that Cherlin's conjecture was correct. The challenge now is to extend this classification to all finite permutation groups. If we are able to do this, then the full force of the model theory developed by Cherlin to understand finite permutation groups will be able to come into play. The proposed research proposes to approach this challenge by utilising the Classification of Finite Simple Groups (CFSG). All finite groups can be broken down into simple groups and, according to CFSG, finite simple groups come in four flavours: they are of prime order, they are alternating, they are a group of Lie type, or they are sporadic. In recent work, two members of the research team developed new techniques that allowed them to completely classify the binary actions of the alternating groups. They were also able to identify a number of new binary actions for the simple groups of Lie type. As a result of this research, we now believe it is possible to classify the binary actions of the finite simple groups. This is the first main aim of the proposed research. This aim will require a focus on the groups of Lie type (a highly theoretical endeavour) and on the sporadic groups (a more computational project). Running parallel to this first main aim is the job of proving a ``reduction theorem'' which connects the binary actions of an arbitrary finite group G to the binary actions of the simple groups of which it is composed. This kind of reduction is a well-established approach to problems in the theory of finite group actions and, indeed, such a reduction was an integral part of the proof of Cherlin's conjecture for finite primitive binary actions mentioned earlier. Nonetheless, the method of proving such reduction theorems varies from case to case, and we expect this to be the more difficult part of the proposed research project. That said, the prize at the end -- a full classification of the binary actions of finite groups, the first level of Cherlin's organising principle -- has the potential to shed a great deal of new light on some of the most ubiquitous objects in mathematics, namely finite groups.
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