Agnes Scott College

Grace Marie Bareis

Imprimitive Substitution Groups of Degree Sixteen
The Ohio State University, 1909


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

a1 a2 a3 a4 a5 a6 a7 a8 b1 b2 b3 b4 b5 b6 b7 b8

for four systems they are

a1 a2 a3 a4 b1 b2 b3 b4 c1 c2 c3 c4 d1 c2 c3 c4

for eight systems they are

a1 a2 b1 b2 c1 c2 d1 d2 e1 e2 f1 f2 g1 g2 h1 h2

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, (a1a2a3a4a5a6a7a8)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 = a1b1.a2b2.a3b3.a4b4.a5b5.a6b6.a7b7.a8b8

(2) Four systems,

   t = a1b1.a12b2.a3b3.a4b4.c1d1.c2d2.c3d3.c4d4
   t1 = a1b1c1.a2b2c2.a3b3c3
   t3 = a1b1.a2b2.a3b3.a4b4

Of these t t1 generate a group simply isomorphic to the alternating group of degree four and t t1 t3 generate one simply isomorphic to the symmetric group of degree 4.

(3) Eight systems,

   t1 = a1b1c1d1e1f1g1.a2b2c2d2e2f2g2
   t2 = a1e1f1d1b1c1g1h1.a2e2f2d2b2c2g2h2
   t3 = b1d1c1g1e1f1.b2d2c2g2e2f2
   t4 = h1a1.b1d1.c1g1.e1f1.h2a2.b2d2.c2g2.e2f2
   t5 = b1g1d1c1.e1f1.b2g2d2c2.e2f2
   t6 = a1b1h1.a2b2h2
   t7 = a1h1.a2h2

Of these t1 t7 generate a group simply isomorphic to the symmetric group of degree eight; t1 t6 to the alternating group; t1 t4 t32 t5 to the primitive group of order 1344; t1 t2 t3 to the 336; t1 t22 t32 to 1681; t1 t4 t32 to 1682; t1 t4 to the 56 group.