Colloquium publications.

142

THE CAMBRIDGE COLLOQUIUM.

this is obviously that it shall be possible to solve the equations
(10) so as to express gi, g2, *, gn by equations analogous to (10)
in terms of gi', g2', *..., g'.
The equations (10) determine an analogous set of equations
for the commutative group G
(11) g = y1 2~ i * gnA (i = 1, 2,., n)
in which
gij = aij + j + * * * + ij
A solution of the equations (10) must correspond to a solution of
the equations (11). But since the elements in (11) are commutative the process of solution is entirely analogous to that of
solving the linear equations,
Xi' -= gUilXl + igi2X2 + * - * + g inXn (i = 1,2,..., n)
in terms of integers. The condition that a unique solution in
integers shall exist is
gil l12... * n
(12) g21 g 22 * /2 n = -4-1.
gn 1 gn2 * * * nn
Hence (12) is a necessary condition that (10) shall be a transformation to a new set of generators of G.
31. If G is to be expressed in terms of the generators g1', g2',
* * *, g', the expressions for gi, g2, '", gn in terms of gi', g2',
gn' must be substituted in the generating relations (1) in order
to obtain a new set of generating relations in terms of gi, g2,, gn. When this set of generating relations is modified by
allowing all elements to be commutative it becomes a new set
of generating relations for G. This set of generating relations
for G could also be obtained by substituting directly the solutions
of (11). This amounts to multiplying the matrix I IYrs I on the
right by a square matrix of n rows and determinant 4 1.
The generating relations (1) can also be modified by replacing