An introduction to the modern theory of equations, by Florian Cajori.

ON SUBSTLTUTIONS

107

The repetition of a substitution s or s-1, r times, is denoted
by sr or s-r. Hence exponents are used here in much the
same way as are integral exponents in algebra.
86. Cyclic Substitutions. If we suppose the letters of the
substitution (aa2... a,,) to be placed in the given order on the
360~
circumference of a circle at equal intervals of, the given
n
substitution is equivalent to a positive rotation of the circle
360~
through. Hence such a substitution is called a cycle, or
a cyclic substitution, or a circular substitution. The product
(abc... ()(xyz... w) is called a substitution of two cycles.
Similarly we have substitutions of three or more cycles. The
substitution ( 34 57 1 6) consists of the two cycles,
\i3 4 5 7 1 2 6)
(1 3 5)(2 4 7 6); for 1 is replaced by 3, 3 by 5, 5 by 1, and
we have one cycle; again, 2 is replaced by 4, 4 by 7, 7 by 6,
6 by 2, and we have the second cycle.
In this manner any substitution can be resolved into cycles
so that no two cycles have a digit in common. This resolution
can be effected in only one way.
A cycle may consist of a single element, say (5). The substitution ( 2 34) 5may also be written (1 3 4)(2)(5), or
(1 3 4)2 5, or (1 3 4).
Ex. 1. Find the cycles of the substitution ( cd fih).
Ex. 2. Verify the relations (acb) (abc) =1, (abc) (abc) = (acb), (ab) (ac)
- (abc), (bc)(acb)=(ac), (bc)(b ()=1, (abc)(acb) 1.
Ex. 3. In which of the following products is the commutative law
obeyed: (abc)(ac), (bc)(acb), (bca)(bac)?

Ex. 4. Write the inverse of (abode).