Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

86

ABSTRACT GROUPS

[CH. III

lished. Even Sylow's theorem appeared two years later, and
a number of other more recent theorems make it possible to
present many subjects in a simpler manner than could be done
at the time when Jordan wrote this Traite. The only other
separate text-books on groups of finite orders which appeared
before the present century are Netto's Substitutionentheorie,
1882, Burnside's Theory of Groups of Finite Order, 1897, and
Bianchi's Lezioni sulla theoria dei gruppi di sostituzioni, 1897
(lithographed). Netto's book was translated into Italian by
G. Battaglini in 1885, and into English by F. N. Cole in 1892.
In the older writings on groups, and in some of the more
recent ones, a set of distinct elements considered with respect
to only one formal law of combination is said to form a group
whenever every element obtained by combining an element with
itself or with any other element of the set is again in the set.
In most cases it is, however, tacitly assumed that these elements
obey the other laws which were imposed in our definition of a
group. If this is not assumed the set may be said to have
" the group property." Cf. M. Bocher, Introduction to Higher
Algebra, 1907, page 82.
Since the beginning of the present century a considerable
number of separate treatises on groups of finite order have
appeared. The literature on this subject has been made more
accessible by the publication of two somewhat extensive bibliographies, viz., The. Constructive Development of Group-Theory
by B. S. Easton, University of Pennsylvania, 1902, and the
"Essai d'une bibliographie sur la theorie des groupes " by C.
Alasia, published in the Rivista di fisica, mathematica e scienze
naturali, Pavia, 1908-10.
The main theorems of the theory of finite groups, together
with numerous historical data relating to their development,
may be found in the Encyclopedie des Sciences Mathematiques,
tome 1, volume 1, page 532, and in the second edition of Pascal's Repertorium der hdheren Mathematik, volume 1, page 168.
A list of remarks on the bearing of the theory of groups,
exhibiting the wide application of this subject, was published
in volume 6 (1914), of the Tohoku Mathematical Journal.