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

122

THEORY OF EQUATIONS

of substitutions is assumed to be finite, this process must come
to an end, and we have the sets
817 827 S7 * * * ' S'n
slt, s2t, sat, *'*, st
Sltl, StWL S3tl... StLtD
ltm,? 2t~7 $8t~, " '**, 8,^tnt.
The total number of substitutions in the group is therefore n
times the number of sets, or (m + 2)n. But (m + 2)n is the
order of the group, and n the order of the sub-group. Hence
the order of the sub-group is a factor of the order of the group.
107. Index of a Sub-group. If n is the order of a group G
and m the order of a sub-group G1, the quotient " is called the
index of G1 under G. Thus the index of an alternating group
under the symmetric group of the same degree is n! 2! 2.
Ex. 1. Give the index of every group of the fifth degree under the
symmetric group.
Ex. 2. Show that a group whose order is prime can have no sub-group
(except the substitution 1).
108. Normal Sub-groups. -If G1 is a sub-group of G, and s
any substitution of G which does not occur in GI, the groups G1
and s-1~Gs are called conjugate sub-groups of G. By the transformation s-lGs, we mean the result obtained by subjecting
every substitution s, of the sub-group G1 to the transformation
s-lsls.
If G, and s-lGls are identical to each other, whatever substitution s is of G, G1 is called a normal sub-group, or a self-conjugate sub-group, or an invariant sub-group of G.
109. Simple Groups. - A simple group is one which has no
normal sub-groups, other than the group consisting of the
identical substitution.