The Ohio State University, 1909

Introduction

Sixteen symbols can be divided in three ways into sets containing an equal number of symbols, viz., into two 8's, four 4's, or eight 2's. In the following list a group is classed under the first systems of intransitivity to which its head belongs. For example a group having both two and four systems of imprimitivity is listed among those having two systems.

In previous lists letters of the alphabet have usually been used as elements. In this list letters with subscripts are used, the letters to distinguish between the systems and the numbers a subscripts to distinguish between elements within the system. The sixteen symbols used differ therefore with the number of systems.

For two systems of imprimitivity they are

*a*_{1} *a*_{2} *a*_{3} *a*_{4} *a*_{5} *a*_{6} *a*_{7} *a*_{8} *b*_{1} *b*_{2} *b*_{3} *b*_{4} *b*_{5} *b*_{6} *b*_{7} *b*_{8}

for four systems they are

*a*_{1} *a*_{2} *a*_{3} *a*_{4} *b*_{1} *b*_{2} *b*_{3} *b*_{4} *c*_{1} *c*_{2} *c*_{3} *c*_{4} *d*_{1} *c*_{2} *c*_{3} *c*_{4}

for eight systems they are

*a*_{1} *a*_{2} *b*_{1} *b*_{2} *c*_{1} *c*_{2} *d*_{1} *d*_{2} *e*_{1} *e*_{2} *f*_{1} *f*_{2} *g*_{1} *g*_{2} *h*_{1} *h*_{2}

The transformations necessary for the final comparison of groups of different systems are very simple.

The transitive constituents of the heads are distinguished by numbering them as they are numbered by Professor G. A. Miller in this "Memoir on the Substitution Groups whose Degree does not Exceed Eight" [American Journal of Mathematics, vol. 21 (1899)]. For example, (*a*_{1}*a*_{2}*a*_{3}*a*_{4}*a*_{5}*a*_{6}*a*_{7}*a*_{8})_{3211} is the eleventh group of order 32 and degree 8 in Professor Miller's list. Each distinct head is written and is followed by substitutions which generate groups that multiplied into the given head produce distinct imprimitive groups. In this list the following notation is used:

(1) Two systems,

*t* = *a*_{1}*b*_{1}.*a*_{2}*b*_{2}.*a*_{3}*b*_{3}.*a*_{4}*b*_{4}.*a*_{5}*b*_{5}.*a*_{6}*b*_{6}.*a*_{7}*b*_{7}.*a*_{8}*b*_{8}

(2) Four systems,

*t* = *a*_{1}*b*_{1}.*a*_{12}*b*_{2}.*a*_{3}*b*_{3}.*a*_{4}*b*_{4}.*c*_{1}*d*_{1}.*c*_{2}*d*_{2}.*c*_{3}*d*_{3}.*c*_{4}*d*_{4}

*t*_{1} = *a*_{1}*b*_{1}*c*_{1}.*a*_{2}*b*_{2}*c*_{2}.*a*_{3}*b*_{3}*c*_{3}

*t*_{3} =
*a*_{1}*b*_{1}.*a*_{2}*b*_{2}.*a*_{3}*b*_{3}.*a*_{4}*b*_{4}

Of these *t* *t*_{1} generate a group simply isomorphic to the alternating group of degree four and *t* *t*_{1} *t*_{3} generate one simply isomorphic to the symmetric group of degree 4.

(3) Eight systems,

*t*_{1} =
*a*_{1}*b*_{1}*c*_{1}*d*_{1}*e*_{1}*f*_{1}*g*_{1}.*a*_{2}*b*_{2}*c*_{2}*d*_{2}*e*_{2}*f*_{2}*g*_{2}

*t*_{2} =
*a*_{1}*e*_{1}*f*_{1}*d*_{1}*b*_{1}*c*_{1}*g*_{1}*h*_{1}.*a*_{2}*e*_{2}*f*_{2}*d*_{2}*b*_{2}*c*_{2}*g*_{2}*h*_{2}

*t*_{3} =
*b*_{1}*d*_{1}*c*_{1}*g*_{1}*e*_{1}*f*_{1}.*b*_{2}*d*_{2}*c*_{2}*g*_{2}*e*_{2}*f*_{2}

*t*_{4} =
*h*_{1}*a*_{1}.*b*_{1}*d*_{1}.*c*_{1}*g*_{1}.*e*_{1}*f*_{1}.*h*_{2}*a*_{2}.*b*_{2}*d*_{2}.*c*_{2}*g*_{2}.*e*_{2}*f*_{2}

*t*_{5} =
*b*_{1}*g*_{1}*d*_{1}*c*_{1}.*e*_{1}*f*_{1}.*b*_{2}*g*_{2}*d*_{2}*c*_{2}.*e*_{2}*f*_{2}

*t*_{6} =
*a*_{1}*b*_{1}*h*_{1}.*a*_{2}*b*_{2}*h*_{2}

*t*_{7} =
*a*_{1}*h*_{1}.*a*_{2}*h*_{2}

Of these *t*_{1} *t*_{7} generate a group simply isomorphic to the symmetric group of degree eight; *t*_{1} *t*_{6} to the alternating group; *t*_{1} *t*_{4} *t*_{3}^{2} *t*_{5} to the primitive group of order 1344; *t*_{1} *t*_{2} *t*_{3} to the 336; *t*_{1} *t*_{2}^{2} *t*_{3}^{2} to 168_{1}; *t*_{1} *t*_{4} *t*_{3}^{2} to 168_{2}; *t*_{1} *t*_{4} to the 56 group.