Colloquium publications.

ANALYSIS SITUS.

139

operations are commutative. The matrix of G is
I 1Yrsl I (r= 1,2,,k;s= 1, 2,, n)
where
'rs = ars + brs + * + jrs.
26. Regarding G as the group of a two-dimensional complex,
the commutative group G can be studied by means of the matrix
E2. For let the oriented 1-cells ail (i = 1, 2, * * *, a,) be denoted
by ai, and also, in the present section, denote the number a, by X.
Then each of the generators gi, g2, '', gn can be expressed in the
form
(2) gi = a(l"ai,2 '" a ix... alr~,a2r~... 2 (i= 1, 2,.. * ).
On substituting these expressions in (1) we find the generating
relations of G expressed in terms of the a's. If the group is set
up in the manner described in ~ 22 each of these relations takes
the form
(3) lI mj 1 = 1
where lj is a set of a's representing a curve from 0 to a point on
the boundary of one of the 2-cells aj2 and mj represents the
boundary of the 2-cell.
On passing to the group G by introducing the condition of
commutativity (3) becomes
(4) mj = 1.
Since mj represents the boundary of the 2-cell aj2 it is expressible
in the form
(5) al elj2e2ij ) ei =1 (j = 1, 2,.., a2)
in which the exponents are the elements of the jth column of the
matrix E2. Hence the generating relations of the group C when
expressed in terms of a1, a2, *., oal, take the form (5) in which the
matrix of the exponents is E2', the matrix obtained by interchanging
the rows and columns of E2.
It is worthy of comment that whereas the group G is defined
in terms of a definite point 0 of C2 (an isomorphic group is