New Proofs of the Simplicity of Every Alternating Group Whose Degree is Not Four
G. A. Miller · Annals of Mathematics · 1925
C. Jordan proved that every alternating group whose degree is not four is simple, Traite des substitutions. 1870, page 66. Several years earlier he had proved that equations belonging to this group are simple, Paris Comptes Bendus, volume 60 (1865), page 773. In what follows we shall establish an elementary theorem from which the simplicity of every alternating group whose degree exceeds 5 and of certain other groups can readily be deduced. This theorem may be stated in the following form: If every subgroup composed of all the substitutions which omit a given letter of a substitution group G whose degree exceeds 4 is at least triply transitive then G must be at least fourfold transitive and cannot involve more invariant subqroups than every one of the given triply transitive stubgroups contains. In particular, when one of these triply transitive subgroups is simple G must be simple. The fact that this theorem does not apply to all the groups of degree 4 is illustrated by the symmetric group of this degree. It is evident that this theorem establishes the simplicity of every alternating group whose degree exceeds 5 provided the simplicity of the alternating group of degree 5 has been established. Hence we shall prove that this group is simple before proving the theorem in question. It may be of interest to note that this special case is also included in a very elementary general theorem relating to the subgroups composed of all the substitutions which omit a given letter. This theorem may be expressed as follows: If a subgroup composed of all the substitutions which omit a given letter of a transitive group is of class p and of degree p + 1, p being an odd prime number, then this transitive group is simple and there is only one such group for a given value of p. It is obvious that the subgroup in question is of degree 2a and involves invariantly the abelian regular group of order 2c and of type (1, 1,1, ...). Since this subgroup is primitive the transitive group G must be of degree p + 2. The group G cannot involve invariantly a regular group of degree p -2 since the subgroup composed of all the substitutions which omit one letter of the holomorph of such a regular group cannot be multiply transitive. For the same reason G cannot involve an invariant subgroup of order (p + 1) (p +2) since such a subgroup would be of class p +I1 and hence would involve a characteristic regular group of order p +2. Since G is 87