1 Abstract and Acknowledgements

I (Alex Brodbelt) am deeply grateful to Christopher Butler for providing the TeX code so I could add to his already incredible exposition of Dickson’s classification of finite subgroups of \(\operatorname{SL}_2(F)\) over an algebraically closed field.

I feel obliged to credit Christopher where it is due.

In order to explain what this paper is about, it is necessary to first define a few of the mathematical concepts which it concerns. A group is a set of objects, called elements, together with a rule, called an operation, which tells us how two elements combine with each other to make a third. Furthermore, to be considered a group it must also satisfy 4 conditions, called axioms. One of which is that the group must be closed under it’s operation. This means that whenever any two elements in the group are combined, the resulting element is also part of the group. The remaining axioms require that the group must also be associative, have an identity element and each element must have an inverse. The way in which the elements in a group act with each other is called the group’s structure. If 2 groups have the same number of elements and share the same structure, then they are regarded as being isomorphic to each other, which essentially means that they equivalent. Many everyday things can be regarded as groups, such as the symmetries of geometrical objects, or the number systems we use.

The set of 2 x 2 matrices whose determinant is equal to 1, together with the operation of ordinary matrix multiplication, forms a group called the special linear group. This is a group because the product of 2 matrices has a determinant equal to the product of the determinants of the 2 matrices, so since 1 x 1 = 1, this new element also belongs to the group, hence the axiom of being closed is satisfied. Furthermore, it is crucial that the entries in the matrices are taken from a specified ring or field. Rings and fields are, like groups, abstract mathematical objects, albeit they satisy even more axioms than groups do. Crucially, rings and fields have both an additive and a multiplicative identity.

This paper focuses on \(\operatorname{SL}_2(F)\), which is the two-dimensional special linear group whose entries are taken from an algebraically closed field. Algebraically closed fields are infinite in size, which means that the resulting special linear group is also infinite. A subgroup of a group is simply a group with the added requirement that each of it’s elements must also belong to the original group. Thus a finite subgroup of \(\operatorname{SL}_2(F)\) is any finite set of elements belonging to this infinite group \(\operatorname{SL}_2(F)\), which satisfy the 4 axioms of being a group.

This paper classifies all the possible structures which a finite subgroup of \(\operatorname{SL}_2(F)\) could have. The result has implications within the study of finite simple groups. This classification was first done by American mathematician Leonard Eugene Dickson in 1901. The purpose of this reformulation is to make it accessible to a wider audience by providing a more detailed explanation at the various stages of the proof.

This paper is a reformulation of Leonard Dickson’s complete classification of the finite subgroups of the two-dimensional special linear group over an arbitrary algebraically closed field, \(\operatorname{SL}_2(F)\). The approach is to construct a class equation of the conjugacy classes of maximal abelian subgroups of an arbitrary finite subgroup of \(\operatorname{SL}_2(F)\). In turn, this leads to only 10 possible classes of structures of this subgroup up to isomorphism.

I would like to take this opportunity to thank my advisor Arne Meurman. This paper would not have been possible without the guidance and insight he gave during our weekly discussions.