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

PREFACE

vA

lectures given by its author at various times and the lecture
notes were frequently changed so as to obviate difficulties which
presented themselves to the students. Special thanks are due
to Dr. E. A. Kircher for assisting the author in preparing these
notes for publication, and to Professor W. A. Manning and Dr.
Josephine E. Burns for valuable suggestions on the printer's
proofs.
Part II, written by H. F. Blichfeldt, seeks to give a more comprehensive outline of the theory of linear groups as developed
up to the present moment than is contained in the published
texts dealing with this phase of group-theory. At the same
time an attempt has been made to present this theory in as simple
a manner as possible, consistent with brevity. Thus, in several
places it has been deemed sufficient to indicate the method of
proof of a general proposition by attending to a concrete case.
From the outset the student is urged to work with the matrix
form of a linear transformation (~ 76). The practice thus
gained is of great advantage throughout Part II; in particular,
the more difficult sections of Chapter XIII will be mastered
readily if the student has a clear mental image of the matrix
form of the regular groups as depicted in ~ 136 (M').
The introductory chapter (IX) and the chapter on binary
groups (X) presuppose only the rudiments of ordinary group
theory as given in ~~ 1-4, 6-9, 22, in addition to a few definitions. By the aid of the Hermitian invariant (~ 92), the
the determination of the binary groups is here made to depend
upon geometrical analysis, entirely with reference to Euclidean
space.
The following two chapters (XI, XII), exclusive of ~~ 116,
125, are based on articles published by the author, mainly in
the Transactions of the American Mathematical Society, 1903-1911.
Certain proofs have been recast and new theorems added. For
the intelligent reading of these chapters and the next following,
the principles of ~~ 1-22, 27, 48 should be well understood.
The somewhat difficult theory of group-characteristics (Ch.
XIII) has been developed along fairly easy lines, differing not
only in arrangement, but also in methods of proof, from pre