144 ' MILLER— SUBSTITUTIOXS ON n LETTERS. [April 20, 



of the substitutions K which are commutative with every substitu- 

 tion of G. In other words, the substitution group according to which 

 the transitive constituents of G are transformed may be intransitive. 

 When this condition is satisfied A' is the direct product of two or 

 more symmetric groups. This suggests a more general infinite 

 system of pairs of amicable intransitive groups than the one men- 

 tioned above : viz., Let G be the direct product of the p groups 

 formed by establishing simple isomorphisms between n^ symmetric 

 groups of degree n^, »„ symmetric groups of degree «„, •••, n 

 symmetric groups of degree ;/ (n^, n^, •••, n being distinct 

 numbers greater than 2), it is clear from what was proved 

 above that K is similar to G and hence G and K are ami- 

 cable intransitive groups. It should be observed that G and K 

 are akuays amicable zvhcncvcr tJicy arc siniilar but that the converse 

 of this theorem is not always true. This more general system of 

 amicable intransitive groups may clearly be constructed by forming 

 the direct product of the p symmetric groups of degrees Hj, jIo, •■-, 

 lip respectively and forming the ;n-sets as regards the subgroups H 

 obtained by forming the direct i)roduct of p symmetric groups of 

 degrees n, — i, n.-. — i, ••-, n — i respectively, one being taken 

 from each of the given symmetric groups, in order. If the /;;-sets 

 thus obtained are multiplied on the right and on the left by all the 

 operators of these sets there clearly results the two systems of 

 amicable intransitive groups in question. 



To obtain a still more general infinite system of amicable intran- 

 sitive groups it should be first observed that the intransitive group 

 formed by establishing a simple isomorphism between in^ symmetric 

 groups of degree n^, written on m^ distinct sets of letters, is amicable 

 with the one obtained by establishing a simple isomorphism between 

 n^ symmetric groups of degree ;;?], written on n^ distinct sets of 

 letters, where ;/,, ui^ > 2. Hence it results that the direct product 

 formed by multi])lying p intransitive groups of degrees n^m^, tunu, 

 "'■> '^p '"p respectively, each being formed in the former of the two 

 ways mentioned above, is amicable with the direct product formed 

 by multiijlying the p groups of the same degrees respectively, but 

 constructed by establishing a simple isomorphism between n^ sym- 



