Colloquium publications.

138

THE CAMBRIDGE COLLOQUIUM.

24. An important though obvious consequence of the last
sections is that any discrete group with a finite number of
generators is the fundamental group of a two-dimensional complex. For, given a group with n generators gi, g2, *.*, gn and
k generating relations, construct a linear graph Ci consisting of
a point 0 and n closed curves having 0 and no other points in
common. Let one of these curves correspond to each generator.
The left hand member of each generating relation denotes a
closed curve on Ci. Introduce a 2-cell (whose boundary is in
general singular) bounded by each of these curves. The result
is a two-dimensional complex having the given group as its
fundamental group.
The Commutative Group G
25. Suppose that a group G is determined by n generators
g1, g2, "', gn and a number k of generating relations. The
latter may be written in the form
gll. g 12. gan. gl' g2 12...gb * * 22.. n= 1,
(1) gla21 g222. *, a2 *g g g21 g22 b gn2n. gi212 g.222 * g 2n = 1
The exponents of the g's are positive or negative integers or zero.
The group is characterized by the matrix of the exponents. This
matrix has k rows and a number of columns which is a multiple
of n. It will be called the matrix of the group.
If the group G is commutative, that is, if gi' gj = gj gi for all
values of i and j, the left member of each expression in (1) can
be written in the form
glarl.g2ar2... g a
Hence in this case the matrix is one of k rows and n columns.
If G is not commutative there is a unique commutative group
G associated with it, namely the group generated by g, g2, *,
gn subject to the conditions (1) and the condition that all the