184 THEORY OF COLLINEATIONS. 
as) ive Oto? 
SS bs 0 : C= 0 b ol, 
came es | @3 3 C2 | 
NG i Seatowatg 
G(AAI) =| 0  0|, G,(AAIA=\0 b 0}, 
0 63 0 0 cs 
It is easy to show by matrix multiplication that both group 
properties hold for each of these matrices, and that each of 
these eight subgroups of G, defined by linear relations is a re- 
duced group. 
220. Necessary Condition of Reducibility. We have thus 
far shown that every group whose defining equations are 
linear is a reducible group. We wish now to find a necessary 
condition of reducibility. A reducible group G’ is one that 
may be transformed by a collineation S, thus SG’S7=G, 
such that for each collineation in G certain elements of its 
matrix are always zero, this being its reduced form. 
Let us first assume that one of the elements in the principal 
diagonal of M, say a,, is always zero. Putting a,=0 and 
a,=0 in equations (10), Art. 172, we have 4,=4@,3,+4571, 
which is not zero; and we have no reducible group. 4, will 
vanish if we assume say a,=0 and y,=0; but then 4, and ¢, 
will not vanish. 4, will vanish if we assume a,=a,=0, or 
3,=y,=0; but then the determinant of M vanishes and we 
have only pseudo-collineations. Hence an element in the 
principal diagonal of M can not be zero for all collineations in 
a group. 
Next let us assume that some element noe in the principal 
diagonal is zero; by suitable interchanges of rows and col- 
umns this element may be brought into the upper right hand 
corner of M; hence without loss of generality we may assume 
c, is always zero. Putting c,=y,=0 in (10) we get ¢,=¢,3,, 
and we have no reducible group unless ¢,=0 or 3,=0. Mak- 
ing c,=¢,=0 and y,=y.=0, we get ¢,=c,=0, and have a 
reducible group; making c,=b,=0 and y,=3,=0 we get 
