Last year when I was preparing for course lectures the work of Ahlfors [1] which establishes that the Bloch constant is at least as large as it appeared to me that the resources of the theory of metrics of negative curvature offered rich possibilities from a function-theoretic point of view. The parallelism between certain properties of subharmonic functions and those of the metrics introduced by Ahlfors [1] is so striking that we are led to ask whether one can introduce a class of metrics including the metrics of Ahlfors for which not only does a Schwarz-Pick-Ahlfors lemma hold, but also requirements of differentiability disappear, as in the modern theory of subharmonic functions.

Nous définirons la structure d’ensemble individualisé au moyen des données suivantes:

—un ensemble non vide A, éléments notés u, v, w,…

—un ensemble infini I, éléments notés a, b, c,… i, j, k,… et appelés individus.

—une application s de I2 dans l’ensemble des applications de A dans lui-même; une fonction telle que s(a, b) sera dite “substitution de a à b”, et lorsque aucune confusion ne sera à craindre, on la notera simplement (a/b), sa valeur pour u ∈ A sera alors notée (a/b)u. Si l’on est en présence de plusieurs individualisations différentes, on pourra préciser en disant que A est s-individualisé sur I.

The contents of this note were reported at a meeting of the Japan Mathematical Society five years ago, but it was not printed. Prof. K. Yano advised me to do so and it was as follows.

Even though the logical structure of any formal deduction can be nicely expressed in a tree-form diagram, it is more practical to write it down in a series of propositions. In each step of inference, we usually deduce a proposition on basis of some foregoing propositions. However, global aspects of mathematical theories show us that this is not always the case. For, in mathematical theories, theorems are usually stated before their proofs. In fact, also in proofs of theorems, it is often practical that we prove propositions after stating them. Accordingly, in our real way of thinking, we arrange propositions going back and forth in the logical order.

Let Q be the set of symbols 1, 2,…, 3 p, where p is an prime number greater than 3. Let be a transitive permutation group on Ω, which is simple and in which the normalizer of a Sylow p-subgroup has order 2 p.

In his address at the International Congress of Mathematicians at Amsterdam [1] Professor R. Brauer proposed a problem of characterizing various groups of even order by the properties of the involutions contained in these groups and he gave characterizations of the general projective linear groups of low dimensions along these lines. The detail of the one-dimensional case has been published in [5], but the two-dimensional case has not appeared yet in detail. His work was followed by Suzuki [7], Feit [6] and Walter [11]. The present paper is a continuation of [73 and discusses a characterization of the two-dimensional projective unitary group over a finite field of characteristic 2. The precise conditions which characterize the group in question will be stated in the first section. The method employed here is similar to the one used in [7]. An application of group characters provides a formula for the order. However a difficulty comes in when one attempts to identify the group. In order to overcome this difficulty we will use a method primarily designed to study a class of doubly transitive permutation groups (cf. [9]). We need also a group theoretical characterization of a class of doubly transitive groups called (ZT)-groups. This is a generalization of a result in [8], and may be of in dependent interest.

The polyhedral group (l, m, n) is defined in [3] by the presentation

E. S. Barnes and I recently constructed a series of positive quadratic forms fN in N = 2n variables (n = 1, 2,…) with relative minima of order for large N. I continue this investigation by determining the minimal vectors of fN and showing that, for its group of automorphs is the Clifford group This suggests a generalization. Replacing by where p is an odd prime, I derive a new series of positive forms in N =(p−1)pn variables (§4). The relative minima are again of order (p fixed, N → ∞ ), the “best” forms being those for p = 3, 5. All forms are eutactic though only those for p = 3,5 are extreme.

Let G be a finite group and let ψ be an (ordinary) irreducible character of a normal subgroup N. If ψ extends to a character of G then ψ is invariant under G, but the converse is false. In section 3 it is shown that if ψ extends coherently to the intermediate groups H for which H/N is elementary, then ψ extends to G. If N is a Hall subgroup, then in order for ψ to extend to G it is sufficient that ψ be invariant under G. This leads to a construction of the characters of G from the characters of N and the characters of the subgroups of G/N in this case.

It may safely said that one of the most important problems in modern algebraic geometry is to elevate theory of abelian functions to the same level as theory of elliptic functions beyond the modern formulation of classical results. Being concerned in such a problem, we feel that one of the serious points is the lack of knowladge on the explicit expressions of abelian varieties and their law of compositions by means of their canonical systems of coordinates: Such expressions correspond to the cubic relation of Weierstrass’ -functions and their addition formulae in theory of elliptic functions.