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

~ 96,

ABELIAN GROUPS

213

By the theory of linear homogeneous equations, a set of
solutions b,...,, not all zero, of these equations, can
always be found if 0 is a root of the characteristic equation
of A (~ 89).
If we now introduce new variables such that yl is one of
these, the group generated by A is reducible, since
(yi)A = yi, (yi)A2 = 02y, etc.,
and therefore
yi = 6y'I
is one of the equations specifying A in its new form. By
Theorem 6, the group generated by A is intransitive, one of
the sets of intransitivity being yi. Let (y2,..., yn) form the
other (temporary) set of intransitivity.
The above process may now be repeated for the set (y2,.., yn)' We determine the linear function 2y2 +.. + CnYn
which is a relative invariant of A, and introduce this function
as one of n-1 new variables to take the place of y2,.., yn.
Continuing thus, the transformation A will finally appear in
the canonical form.
96. Theorem 8. In any given abelian group K (~ 26) of
linear transformations, such new variables may be introduced
that all the transformations of K will simultaneously have the
canonical form.
Proof. If the group contains only similarity-transformations, the theorem is self-evident. Hence we assume in K
a transformation S which is not a similarity-transformation.
Let the variables of the group be chosen such that S appears
in the canonical form
S=(oa,., a; o02,..., 02;.. *; On, ~.., On),
the variables being arranged so that those having the same
multipliers are grouped together. Let there be a variables
xi,..., xa having the multiplier ac; b variables x+,..
Xa+b having the multiplier a2, etc.
Now let T be any transformation in K. Since TS=ST,