In a previous paper [3] we introduced indicator sets in order to facilitate the study of partial spreads and spreads in ∑ = PG(3, q). The idea enabled us to disprove a conjecture in the literature by constructing a spread that contained no regulus. More recently there have been some notable advances in the theory of spreads. One such advance is Denniston's theorem that packings of spreads always exist in ∑. This result can be very nicely interpreted in terms of indicator sets [1], Another notable advance is the result on subregular spreads due to W. F. Orr [4; 5] which is discussed below. In this note we develop further some ideas on indicator sets in [3] and then use these results to give an alternative proof of this result of Orr.

There are several notions of positive definiteness for functions on topological groups, the two of which are: Bochner type positive definite functions and integrally positive definite functions. The class P(F) of positive definite functions for the class F can be defined more generally and it is interesting to observe that a change in F produces a different class P(F) of positive definite functions. The purpose of this paper is to study the functions in P(LP(G)) which are positive definite for the class LP(G) (1 ≦ p < ∞), where G is a compact or locally compact group. The relevant information about the class P(F) can be found in [1; 2; 3 and 8].

With , denote by Λk the class of functions ƒ of the form

which are analytic in and which map y onto the complement of a domain with boundary rotation at most . It is known [2] that ƒ ∈ Λk if and only if there exist regular starlike functions s1 and s2, with

such that

A Steiner triple system (briefly STS) is a pair where S is a finite set and is a collection of 3-subsets of S (called triples) such that every pair of distinct elements of S belongs to exactly one triple of . The number |S| is called the order of . It is well-known that there is an STS of order if and only if or 3 (mod 6). Therefore in saying that a certain property concerning STS is true for all it is understood that or 3 (mod 6). An STS of order v will sometimes be denoted by .

In this paper we compare the properties separability and modularity for field extensions. Let be fields of characteristic . K is separable over if K and are linearly disjoint over . K is modular over if K and are linearly disjoint over their intersection for all n > 0. The latter definition is due to Sweedler [12] and is important particularly for Galois theories of purely inseparable extensions [2; 3; 4; 7].