125]
123
125.
ON THE THEORY OF GROUPS, AS DEPENDING ON THE
SYMBOLIC EQUATION 9 n =l.
[From the Philosophical Magazine, vol. VII. (1854), pp. 40—47.]
Let 9 be a symbol of operation, which may, if we please, have for its operand,
not a single quantity x, but a system (x, y, ...), so that
0(x, y, ...) = {x', y', ...),
where x', y', ... are any functions whatever of x, y, ..., it is not even necessary that
x', y, ... should be the same in number with x, y, .... In particular x', y', &c. may
represent a permutation of x, y, &c., 6 is in this case what is termed a substitution;
and if, instead of a set x, y, ..., the operand is a single quantity x, so that dx = x' =fx,
6 is an ordinary functional symbol. It is not necessary (even if this could be done)
to attach any meaning to a symbol such as 9 + cf>, or to the symbol 0, nor con
sequently to an equation such as 6 = 0, or 0 ± (f> = 0; but the symbol 1 will naturally
denote an operation which (either generally or in regard to the particular operand)
leaves the operand unaltered, and the equation 6 = <£ will denote that the operation
6 is (either generally or in regard to the particular operand) equivalent to </>, and
of course 9=1 will in like manner denote the equivalence of the operation 9 to the
operation 1. A symbol 0(f) denotes the compound operation, the performance of which
is equivalent to the performance, first of the operation <£, and then of the operation
0; 9(f) is of course in general different from cf)0. But the symbols 9, </>,... are in
general such that 9. cf)x = 9<f>. x> &c., so that 9(f>x> & c - have a definite signi
fication independent of the particular mode of compounding the symbols; this will
be the case even if the functional operations involved in the symbols 9, (f>, &c.
contain parameters such as the quaternion imaginaries i, j, k; but not if these
functional operations contain parameters such as the imaginaries which enter into the
theory of octaves, &c., and for which, e.g. a. ¡3<y is something different from a/3. y,
16—2
