NATURE 
481 
THURSDAY, MARCH 
to 
5, 1897. 
GALOISIAN ALGEBRA. 
Lehrbuch dey Algebra. Von Heinrich Weber. Zweiter 
Band. Pp. xvit+ 796. (Braunschweig: Vieweg und 
Sohn, 1896.) 
N one of Mrs. Barbauld’s stories a domestic fairy, 
with one touch of her wand, transforms a tangled 
heap of parti-coloured silk into an orderly array of neatly 
wound skeins. Not unlike this is the effect of group- 
theory upon mathematical analysis; and it has been 
truly said that, for some time to come, the progress of 
analysis will be approximately estimated by the advance 
in our knowledge of the constitution of groups. 
We have, therefore, good reason to be grateful to Prof. 
Weber for the very clear and masterly exposition of 
group-theory which is contained in the first three books 
of his second volume. It may be that we have read it 
at the psychological moment; in any case, it seems to 
us the clearest and most interesting account of the 
subject that we have seen. 
In the spirit of Cayley’s @c/um that a group is defined 
by the laws of combination of its symbols, the author 
begins by a perfectly abstract definition of a group, and 
develops the theory of its normal and other divisors, 
the composition of its parts, and a series of important 
theorems on the decomposition of a group and _ its 
associated indices. The first chapter concludes with a 
further and more general discussion of metacyclic 
groups, already introduced in Vol. i. 
Chapter ii., on Abelian groups, is substantially a 
revised and improved version of the author’s well-known 
memoir in the Acta Mathematica. The discussion of 
the characters of an Abelian group, in particular, seems 
to us much more easy to understand than the corre- 
sponding part of the original memoir. The most im- 
portant results in this chapter are the existence of a 
basis; the isomorphism of groups with the same in- 
variants ; the fact that to every divisor of an Abelian 
group, of index 7, corresponds a set of exactly 7 
characters, which for all elements of the divisor have 
the common value 1, while for every other element of 
the group at least one of the characters has a value 
different from unity ; and, finally, that every divisor is 
associated with a definite reciprocal group whose degree 
is equal to the index of the divisor. 
The next chapter, which again reproduces, in great 
measure, Prof. Weber's original memoir, contains a com- 
plete discussion of the groups of a cyclotomic corpus. 
It is impossible to give a brief analysis of this very 
important chapter: it must suffice to say that a definite 
algorithm is given for determining all cyclotomic corpora 
which correspond to a given set of invariants, and for 
constructing the associated cyclotomic periods. Chapter 
iv. contains applications of the general theory to cubic 
and biquadratic corpora, and a proof that all Abelian 
corpora of the third and fourth degrees are cyclotomic. 
In other words, the roots of an Abelian cubic or biquad- 
ratic equation with rational integral coefficients may 
1 The first volume of this work was reviewed in NaTuRE of November 12, 
1896 (pp. 25-28). 
NO. 1430, VOL. 55] 
always be expressed as rational and integral functions of 
roots of unity. This is a special case of a very remarkable 
theorem of Kronecker’s, first proved by Prof. Weber, 
and demonstrated later on in the present work. 
Chapter v. contains a further discussion of groups in 
general, and brings the reader fairly abreast of con- 
temporary research. The very real advance which has 
been made in this subject in recent years may be said 
to date from the publication of Sylow’s fundamental 
theorem that if # is the degree of a group and #2 a 
power of a prime which divides 7, the group contains a 
divisor of degree f*. A very simple inductive proof 
(after Frobenius) is given in this chapter; and this is 
followed by a series of propositions, hardly less im- 
portant, and more or less depending upon it. Then we 
have a remarkable theorem, due to Frobenius, that if the 
degree of a group is not divisible by a square, it must 
be metacyclic; and the other one, also discovered by 
Frobenius, that every group whose degree is f*g, where 
é and g are different primes, is metacyclic. These 
theorems dispose of most groups whose degrees do not 
exceed 100: the rest are separately discussed in § 34, 
where references are given to the recent papers of Cole, 
Holder, and Moore. It may be remarked that English 
mathematicians are devoting a good deal of attention 
to group-theory at present: reference might well have 
been made to the work of Askwith and Burnside. The 
last article of this chapter contains a proof of the 
theorem that the permutation-group of 7 letters contains 
no transitive and primitive divisor of index not exceed- 
ing 7, except the alternate group of index 2; a further 
exception being made for 7=4, and for 77=6 respectively. 
Although the proof given is, of course, perfectly sound, 
it does not seem the truly ideal one; and it may very 
well happen that in this, as in other similar cases, a more 
appropriate demonstration will be ultimately discovered. 
Book IJ. deals with linear groups, and in particular 
with the polyhedral and congruence groups with which 
the researches of Klein have made us so familiar. The 
polyhedral groups are exhibited in an analytical form, 
which makes it comparatively easy to discern their sub- 
groups ; the proof that, besides the polyhedral groups, 
there are no other finite groups of the same type is after 
the manner of Gordon, and is remarkably simple in 
character. The decomposition of congruential groups 
(to a prime modulus) is effected very easily with the 
help of Galoisian imaginaries. The whole book may 
be profitably compared with the corresponding part of 
Klein’s “‘ Modulfunctionen,” which, of course, traverses 
much the same ground. 
Book III. contains various interesting applications of 
group-theory. The first chapter is on metacyclic equa- 
tions, especially those of degree #*, where # is a prime. 
It is shown that the Galoisian group of a primitive 
irreducible metacyclic equation of degree #* is isomorphic 
with a linear congruence group (mod. /) of a variables ; 
this group is compounded of a metacyclic group, 
isomorphous with 
2: = 2: + a; (mod. £), 
and a homogeneous congruence group. Thus the problem 
of finding all such metacyclic equations is reduced to 
that of finding all the metacyclic divisors of the homo- 
geneous congruence group. With the help of these 
y 
