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

SUBSTITUTION-GROUPS

121

Ex. 1. By inspection, find sub-groups of
1, (xy)(zw), (xz)( (y), (xw) (z).
Ex. 2. How many sub-groups has G24(4)? See ~ 104.
Ex. 3. How many sub-groups has G12(4)?
Ex. 4. What sub-groups has (abcde)lo? (abc) all (de)? (abcde) all?
106. Theorem. The o'rder of a sub-groulp is a factor of the
order of the group to which it belongs.
Let the substitutions of the sub-group be s, s2, s,3,.**, s,,
and let t be any substitution of the group which does not occur
in the sub-group. Then, by the definition of a group, we
know that
Sit, s2t3, S3t *, set, I
are all substitutions belonging to the group, but none of them
belong to the sub-group; for suppose stt =,, then
81- 8ls = S1 Lt = t.
Since s,~- is a substitution of the sub-group (see Ex. 5, ~ 96), it
follows that its product with s,., namely t, belongs to the subgroup - which is contrary to supposition.
Moreover, the new substitutions in I are all distinct; for sup-,
pose s2t = st, then it would follow that s2- = s.
If the substitutions in I do not exhaust the substitutions in
the group not belonging to the sub-group, then suppose the
substitution tl is among those left over. Then
sMtl, s2tl, s3tl, **, st, I:
are distinct substitutions of the group not found in the list
s1, S;, Sn, for reasons just mentioned; nor are they found in T;
for suppose s1t = sit, then t = sj-lst = s,.t, which is some substitution in I, a conclusion contrary to the assumption concerning t,. Continuing in this way, the substitutions of the group
are divided into sets of n substitutions 'each. As the number