An introduction to the modern theory of equations, by Florian Cajori.

SUBSTITUTI ON-GROUPS

117

the digits are divided into the two sets 1, 3, 5 and 2, 4, 6, then
we notice that each of the three substitutions (1 2 3 4 5 6),
(1 4)(2 5)(3 6), and (1 6 5 4 3 2) replaces the digits of one set
by the digits of the other set, while each of the two substitutions (1 3 5)(2 4 6), (1 5 3)(2 6 4) simply interchanges the
digits of one set among themselves. This group is called
imprimitive.
A transitive group is called imprimitive when its elements can
be divided into sets of an equal number of distinct elements, so
that every substitution either replaces all the elements of one
set by all the elements of another, or simply interchanges
the elements of one set among themselves. Otherwise it is
primitive. Example of a primitive group:
1, (1 2 3), (1 3 2).
There are three imprimitive groups of degree four, twelve of
degree six, and no imprimitive groups of degree two, three, and
five.
Ex. 1. Show that no group whose degree is a prime number can be
imprimitive.
104. List of Groups of Degree Two, Three, Four, and Five. We
give here a list of the groups of the first five degrees, omitting
only the group 1. By GqP) we mean a group of the degree
and order q. We give also the notation for groups used by
Cayley and others. In their notation the symmetric group of
degree four is designated by (abed) all; eyc means "cyclic"
substitution; pos means " positive " or even substitution. For
a list of all groups whose degree does not exceed eight, see
Am. Jour. of Math., Vol. 21 (1899), p. 326. In the list of
groups of degree n, we give only those which actually involve n
letters. But it must be understood that any group involving less
than n letters may be taken as an intransitive group of the nth
degree. For instance, G(2)- 1, (ab) may be written as a group
of the third degree, thus: 1, (ab)(c).