On the Composition of Group-Characteristics
William R. Burnside · Proceedings of the London Mathematical Society · 1901
Characteristics " (Prnc.Lond.Math.Soc, Vol.XXXITI., p. 146) I have already given a short account of what Herr "Frobenitis had called their " composition."*In the present communication I consider in greater detail the system of relations of the form which indicate how the various irreducible representations of a group combine among themselves.The main result arrived at, which is, I believe, new, indicates how from the complete system of relations of the above form the existence of each self-conjugate sub-group which the group possesses may be deduced.If G is a group with r sets of conjugate operations, the r distinct representations of the group as an irreducible group of linear substitutions will be denoted by (?" G t , ..., G r .Of these G^ will always be used to denote that representation in which each operation corresponds to identity.In G ( the characteristics of the conjugate sets are .t .and the number of variables is xj.I recall .briefly the process on which the so-called composition depends.Let Gi and Gj be actually set up as groups of linear substitutions in the two distinct sets of variables a-,, a\,, ..., ay ; and y u y v ..., y i.To every operation of G there will then correspond a definite linear substitution on the \\ x{ products of the a;'s and ?/'s, so that G is thus * The process made use of by Herr Frobenius to obtain the composition of characteristics is given in a slightly different connection by M. Jordan {Traite dcs