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

246 THE LINEAR GROUPS IN THREE VARIABLES [CH. XII

therefore form a group P. The order of this group must be
a power of p, since it contains no transformations whose order
differs from p or 1. Moreover, P is invariant in G, since an
operator of order p is transformed into one of order p. Hence, G
has an invariant subgroup P of order pi. But this subgroup
is abelian (~ 108, Corollary) and therefore G is intransitive or
imprimitive (~ 108, Lemma).
Notation 2. A quantity N, which is the sum of a certain
number of roots of unity, in which every root E, is replaced by
1, but in which none of the other changes indicated in 2~ are
carried out, will be denoted by ND. If N=0, then N,=0O
(mod p).
118. Theorem 14. If a group G contains a transformation
S of order p24, p being a prime >2, then there is an invariant
subgroup Hp in G (not excluding the possibility G=Hp) which
contains SD. Any transformation in H,, say T, has the property
expressed by the following congruence:
(11) [V] [ VT], (mod p),
V being any transformation of G.
In the case p = 2 the group G contains an invariant subgroup
H, if the order of S is p3, and SI' will belong to Hp; also if
S = (-1, i, i), in which case S2 will belong to Hp.
The proof follows the plan of that of the previous theorem.
If p> 2, we write S in canonical form, and construct the products
VS, VS2, VR, where R denotes SD. Assuming that the three
multipliers of S are all distinct, we obtain an equation corresponding to (6) in 1~:
[VR] +K[V] +L[ VS] +M[VS2] =.
However, the changes indicated in 2~ are not carried out except
that 1 is put for every root ep whose index is a power of p (cf.
Notation 2 above). The coefficients L, M become multiples
of p by this change, and we find K_=- (mod p). Hence
finally,
[VR], - [V], 0 (mod p).
Now consider all the conjugates Ri,..., R, to R within G.
They generate an invariant subgroup H, (~ 14, Ex. 6), and