### Primitive Groups with a Determination of the Primitive Groups of Degree 20

American Journal of Mathematics, Vol. 34, No. 1. (January, 1912), pp. 1-20

Introduction

Substitution groups were first considered in connection with the solution of algebraic equations. The earliest work that devotes considerable attention to these groups (Ruffini, *Teoria generale delle equazioni*, Bologna 1799) divides the substitution groups, or permutations, as Ruffini calls them, into two main classes, viz., the cyclic and non-cyclic permutations. The non-cyclic permutations are again divided into permutations of the first, second, and third kind, which correspond to what are now known as intransitive, imprimitive, and primitive groups.

The primitive group notion was emphasized later by Galois, who defined primitive groups in connection with primitive equations and who was the first to show the close relation which exists between the theory of substitution groups and the solution of algebraic equations. From his time mathematicians have recognized the fact that the determination of all the substitution groups of a given degree is a fundamental problem of algebra. Since the determination of intransitive groups is based on a knowledge of the transitive groups of lower degree and the determination of imprimitive groups requires a knowledge of primitive groups of lower degree, it is evident that the determination of the primitive groups of different degrees is important for the solution of this fundamental problem. Numerous theorems have been published which aid in this determination, and the enumeration of all the primitive groups of a given degree has been carried through degree 19.

The first part of the present paper contains a number of theorems concerning the determination of simply transitive primitive groups. In Part II, the primitive groups of degree 20 are determined, while in Part III certain primitive groups whose maximal subgroup contains a transitive constituent of prime order are considered.

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