>
> Julie Tolmie wrote:
>
> -> When did cyclic groups come to be defined by index notation (as
> -> powers of a generator) and by whom?
>
> I think around the turn of the century by Harold Hilton, although
> I am unable to cite the work. Perhaps another member of the list
> can supply further information.
>
Hilton's book Finite Groups (1908) contains the usual definition of a
cyclic group as one consisting of the powers of a single element, with
the usual notation. This is preceded by Burnside's book (1st ed. 1897),
except that Burnside uses the term Cyclical. Much earlier Cayley, in his
paper On the Theory of Groups as depending on the symbolical equation
\theta^n = 1, Phil. Mag. 7 (1854), 40-47 (= no. 125 in Cayley's Collected
Works, v.2 (1889), 123-130), mentions groups all of whose elements are of
the form 1, \alpha, ..., \alpha^{n-1}. He says that a group of prime
order is always of this form, while one of composite order may or may not
be of such form. He also says that for all orders, such a group "is in
every respect analogous to the system of the roots of the ordinary
binomial equation x^n - 1 = 0".
Parts 2 and 3 of the same paper appeared in Phil. Mag. 7, 408-409 (Coll.
works no. 126, v.2, 131-132), and Phil. Mag. 18 (1859), 34-37 (Coll.
works 243, v.4 (1891), 88-91). They also refer occasionally to groups of
the above form. The term "cyclical group" occurs in the following two later
papers, though it is never defined, but the meaning is clear from the
context:
On the substitution groups for two, three, four, five, six, seven, and
eight letters, Quart. Math. J. 25 (1891), 71-88, 137-155 (Coll. Works
918, v.13 (1897), 117-149).
Illustrations of Sylow's theorem on groups, Messenger Math. 23 (1894),
59-62 (Coll. Works 957, v.13, 530-533).
Avinoam Mann