Sets of Operations in Relation to Groups of Finite Order. 319 



" Sets of Operations in Eelation to Groups of Finite Order." By 

 A. N. WHITEHEAD, M.A., Fellow of Trinity College. Cam- 

 bridge. Communicated by Professor A. R FORSYTE, F.E.S. 

 Beceived January 19, Head February 2, 1899. 



(Abstract.) 



Introduction. 



The present paper is concerned with the Theory of Groups of Finite 

 Orders. The more general object of the paper is to place this theory 

 in relation to a special algebra of the type considered in the general 

 theory of Universal Algebra. This special algebra, which may be 

 called the Algebra of Groups of Finite Order, has many affinities to 

 the Algebra of Symbolic Logic; and a comparison of it with this 

 algebra is given in the last section of this paper. 



Mathematicians are accustomed in the study of quaternions to the 

 idea of a vector symbol being considered from two points of view 

 according to circumstances, namely, either as a geometrical entity, or 

 as a symbol expressive of the operation of modifying some geometrical 

 entity into another geometrical entity. Now from the point of view 

 of this paper it is natural to abandon the idea of a group of N opera- 

 tions S , S 15 ...... S x-1 on some unspecified object, as being an idea which, 



however vaguely, appertains to a special interpretation of the symbols. 

 The N symbols S , S 15 ...... S N-1 are to be considered, as in the similar 



case of quaternions, primarily as N distinct objects. When two of 

 these objects are multiplied, as in S q S r , then a third object of the 

 group, such as S^, is produced ; and in reference to this multiplication 

 one of the symbols, say S g , may be looked on as an operation on S r 

 modifying it into S^. But this is not the sense in which the symbols 

 are usually called operations in the Theory of Groups. However, in 

 order not to disturb the well understood nomenclature of the subject, 

 the N objects S , S x , ...... S^j will always be called the fundamental 



operations, or, more shortly, the operations. But the word operation 

 can simply be regarded as a name for the objects represented by these 

 N symbols. 



These N symbols are considered to be capable of addition according 

 to the law 



This is the well known law of addition in Symbolic Logic, and the 

 introduction of numerical symbols as factors is thereby avoided. 



The sum of a selection of the N fundamental operations, such as 

 Sp + S g + S r + S^, is called a set. If a set obeys certain special conditions 

 it is called a group. The sum of the whole number (N) of fundamental 



