Our aim in this paper is to consolidate and extend some of the notions in (1; 2; 5; 6) concerning free planes in order to facilitate the study of their collineation groups. An upper bound mn for the orders of the finite subroups of Gn will be established, where Gn is the collineation group of the free plane ƒn of rank n + 4. In the process, a result of (6) will be generalized. Indeed, mn will be shown to be the best upper bound for all n ≠ 5.

By a d-polytope we shall mean a d-dimensional convex polytope. We shall denote a j-dimensional face (or j-face) of a polytope by Fj. Every d-polytope P has proper j-faces for 0 ≦j ≦d — 1 and we shall also say that P is a d-face of itself. Observe that every face of a polytope is again a polytope. The collection of all convex polytopes shall be denoted by .

Kakutani (2) has proved a very general theorem, giving necessary and sufficient conditions for two infinite product measures to be mutually absolutely continuous. To formulate Kakutani's result, let us first recall that a measurable space is a pair (E, B), where B denotes a Borel field (also called σ-ring) of subsets of E, and a measure m on this space is a countably additive set function on B (see Halmos (1)).

In the present paper, we shall prove the following result.

Theorem A. The groups PSL(2, p) can be presented by the following system of generators and relations:

1.1

This theorem considerably improves earlier results of Bussey, Frasch, and Todd (cf. 2, pp. 93-96). The presentation of Frasch reads as follows:

1.2

In this paper we give some results on the extensions of Lie algebras, with emphasis on the case of prime characteristic, although part of the paper is also of interest at characteristic 0. An extension of a Lie algebra L is a pair (E, π), where £ is a Lie algebra and π is a homomorphism of E onto L. The kernel K of the extension is ker π.

In (4), Seitz characterized those finite groups which have exactly one non-linear irreducible character (over the complex numbers). In this paper we are concerned with the general question of what can be deduced about a finite group G if the number of its non-linear irreducible characters m(G) is given. In particular, does the assumption that m(G) is in some sense small when compared with the order |G| impose any restrictions on the structure of G?

By a Reidemeister plane we mean a projective plane having the property that every ternary ring coordinatizing it has associative addition. Finite Reidemeister planes have been investigated by Gleason (2), Liineburg (6), and Kegel and Luneburg (4). In the first paper, Gleason proved that if the order of the plane is a prime power, then it is Desarguesian. Luneburg showed that this is still true if the order is not 60. In the third paper, this last restriction is removed. For infinite planes, the only result is the following theorem due to Pickert (7, p. 301).

Let G be a finite group and let A be a finite solvable operator group on G. Suppose that A and G have relatively prime orders. Let T be the fixed-point subgroup of G with respect to A. We say that A fixes a complex character ζ of G if ζ (gα) = ζ (g) for all g ∈ G and α ϵ A. Our aim in this paper is to define a one-to-one correspondence between the irreducible characters of T and those irreducible characters of G that are fixed by A, and to prove some properties of this correspondence that were mentioned in (8).

In (5), the following theorem was proved in a very general setting:

(1) An additive set function is non-negative whenever its lower derivative is non-negative.

For a continuous additive function of intervals, theorem (1) can be improved as follows:

(2) A continuous additive set function is non-negative whenever its lower derivative is non-negative except, perhaps, on a countable set.

Throughout this paper, G will denote a compact (Hausdorff) topological group with identity e. When G is abelian, there is no difficulty in relating the group multiplication in G to the multiplication in the dual of G since characters are homomorphisms with respect to pointwise multiplication and pointwise multiplication of characters yields another character. However, in the non-abelian case, there are two multiplications associated with the dual of G: (1) representations are homomorphisms with respect to composition multiplication, and (2) the tensor product of representations yields another representation.

Undirected graphs and graph homomorphisms as introduced by Sabidussi (6, p. 386), form a category that admits a categorical product. For the category of graphs and full graph homomorphisms, the categorical product was introduced by Čulik (1) under the name cardinal product. It was independently defined by Weichsel (8) who called it the Kronecker product and investigated the connectedness of products of finitely many factors. Hedetniemi (4) was the first to make use of the fact that the cardinal product is categorical.

Let M be a smooth, closed, simply connected manifold of dimension greater than 5. Let T be an involution on M with a positive, finite number of fixed points. Our aim in this paper is to prove the following theorem (which is somewhat like that of Wasserman (7)).

Let X be a space. We are interested in the homotopy-commutativity of the loop-space ΩX and the suspension ΣX, that is, in the question whether or not nil X ≦ 1, conil X ≦ 1, respectively. Let c: ΩX× ΩX ⟶ ΩX, c': ΣX ⟶ ΣX V ΣX be the commutator and co-commutator maps, respectively.