Let G be a finite group of odd order with an automorphism ω of order 2. The Feit-Thompson theorem implies that G is soluble and this is assumed throughout the paper. Let Gω denote the subgroup of G consisting of those elements fixed by ω. If F(G) denotes the Fitting subgroup of G then the upper Fitting series of G is defined by F1(G) = F(G) and Fr+1(G) = the inverse image in G of F(G/Fr(G)). G(r) denotes the rth derived group of G. The principal result of this paper may now be stated as follows: THEOREM 1. Let G be a group of odd order with an automorphism ω of order 2. Suppose that Gω is nilpotent, and that G(r)ω = 1. Then G(r) is nilpotent and G = F3(G).

In this paper we give some algorithms for determining αw(T) and βw(T), the generalized internal and external stability numbers respectively, of a finite directed tree graph T whose nodes are weighted by a function w. We define αw(T) and βw in section 2. When w gives every node of T the weight 1 then αw(T) = α(T) and βw(T) = β(T) where α(T) and β(T) are the usual stability numbers.

If we think of the input to a queueing system as arising from some process and depending on the history of that process, we might well expect the duration of inter-arrival intervals to depend mostly on the recent history and to a much smaller extent on that which is more remote.

The concept of critical group was introduced by D. C. Cross (as reported by G. Higman in [5]): a finite group is called critical if it is not contained in the variety generated by its proper factors. (The factors of a group G are the groups H/K where K H ≦ G, and H/K is a proper factor of G unless H = G and K =1). Some investigations concerning finite groups and varieties depend on the investigator's ability to decide whether a given group is critical or not. (For instance, one of the crucial points in the important paper [9] of Sheila Oates and M. B. Powell is a necessary condition of criticality: their Lemma 2.4.2.) An obvious necessary condition is that the group should have only one minimal normal subgroup: the group is then called monolithic, and the minimal normal subgroup its monolith. This is, however, far from being a sufficient condition, and it is the purpose of the present paper to give some sufficient conditions for the criticality of monolithic groups. (We consider the trivial group neither monolithic nor critical.) The basis of our results is an analysis of the following situation.

A topology on a set X is defined by specifying a family of its subsets which has the properties (i) arbitrary set intersections of members of belong to , (ii) finite set unions of members of belong to and (iii) the empty set □ and the set X each belong to . The members of are called the closed subsets of X. If X is any subset of X then denotes the closure of X, that is, the set intersection of all closed subsets which contain X, however when X = {x} contains one point only we will denote by . The pair (X, ) is called a topological space or, in what follows, a T-space. By a T-lattice we mean a complete distributive lattice of sets in which arbitrary g.l.b. means arbitrary set intersection, finite l.u.b. means finite set union and which contains the empty set □ It is well-known, for example Birkhoff [1], that if (X, ) is a T-space and the members of are partially ordered by set inclusion then is a T-lattice.

It has been shown that the assumption of a specified cosmological substratum leads to the Lorentz-equivalencc of observers whose uniform motion in the substratum lies along a common straight line. The result is generalised for any pair of unaccelerated observers thus confirming that only the relative velocity of such observers is relevant in relating their observations of an event.

Journal of the Australian Mathematical Society 4 (1964), 452–453

The second paragraph should be deleted. The alleged commutator identity (3) is false and is certainly not due to Philip Hall. The correct form is

as Dr. N. D. Gupta of Canberra has pointed out to me. According to Professor B. H. Neumann, this identity appeared in his (Professor Neumann's) thesis.

Nevertheless the theorem is valid and the proof given is correct.

Let A be an associative algebra over the field F. We denote by ℒ(A) the lattice of all subalgebras of A. By an ℒ-isomorphism (lattice isomorphism) of the algebra A onto an algebra B over the same field, we mean an isomorphism of ℒ(A) onto ℒ(B). We investigate the extent to which the algebra B is determined by the assumption that it is ℒ-isomorphic to a given algebra A. In this paper, we are mainly concerned with the case in which A is a finite- dimensional semi-simple algebra.

Let S be a compact topological semigroup, and let be the collection of all normalized non-negative Borel measures on S. It is well-known that , under convolution and the topology induced by the weak-star topology on the dual of the Benach space C(S) of all complex valued continuous functions on S, forms a compact topological semigroup which is known as the convolution semigroup of measures (see for instance, Glicksberg [3], Collins [1], Schwarz [5] and the author [4]). [1], Schwarz [5] and the author [4]). Professor A. D. Wallace asked if the process of forming the convolution semigroup of measures might be generalized to a more general class of set functions, the so-called “modular functions.” The purpose of the present note is to settle this question in the affirmative under a slight restriction. Before we are able to state the Wallace problem precisely, some preliminaries are necessary.

Journal of the Australian Mathematical Society 5 (1965), 169–195

Theorem 9 (p. 183) should read: The indices of any closed curve inD have the form

n+ = hp/s, n− = hq/s

for some integer h, where s (1 ≦ s ≦ n) is the largest number of identical blocks into which the signature σ can be partitioned. Moreover, for any integer h there exists a closed curve in d with these indices.

In a recent study of generalized transfinite dimeters [4, 5] some geometric extremal problems were encountered. These form the subject matter of this note.