Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

202

GROUPS OF LINEAR TRANSFORMATIONS [uH. IX

and let the ratios xi/x,,..., x,_-/xn be denoted by yl,, 3n — respectively. Then from A we get
a ly'l+ *.. +a, n-lyn-i+asn (s= 2... 1)
anly 1+. +ann-_ly n-l+a, n
To a similarity-transformation here corresponds the identity
yS=yI (= l, 2,..., n-),
and we see that the linear fractional group is simply isomorphic
with the corresponding collineation-group and may be regarded
as its equivalent.
EXERCISES
1. Prove that the similarity-transformations contained in a linear
group G form a subgroup which is invariant in G.
2. Prove that the determinant of a linear transformation belonging
to a finite group is a root of unity (cf. ~ 116).
3. Among the determinants of the transformations of a group G of
order g let there be one which is a root of unity whose index is the power of
a prime p (~ 116). Prove that if all those transformations be eliminated
whose determinants contain as a factor a root whose index is the highest
power of p occurring among such indices, then will the remaining transformations form an invariant subgroup of G of order g/p.
In particular, prove that the transformations of determinant unity
form an invariant subgroup of G.
4. Let T be one of the transformations of the group in the last exercise whose determinant contains a factor e of index pa, and assume that
p is relatively prime to the number of variables n. Then we can always
find a root of unity, say s,, of index pa, whose nth power is - 1. If now all
the elements of the matrix of T be multiplied by A/, the determinant of the
new transformation T' will no longer contain as a factor a root whose index
is a power of p.
(For instance, let T= D 0
(For instance, let T= T=co. Here and T'=)
Now prove that if all the transformations be modified in this manner
we shall obtain a group G' which is isomorphic with G.
Evidently, to a possible similarity-transformation ( a,..., a)
of G whose multipliers a are roots of unity of index a power of p will
correspond the identity (1, 1,..., 1) of G'. In such a case therefore
the order. of G' will be that of G divided by a power of p.
5. Construct the group Ga corresponding to the linear group of order
8 given in ~ 85.