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Isaacs and Seitz conjectured that the derived length of a finite solvable group $G$ is bounded by the cardinality of the set of all irreducible character degrees of $G$. We prove that the conjecture holds for $G$ if the degrees of nonlinear monolithic characters of $G$ having the same kernels are distinct. Also, we show that the conjecture is true when $G$ has at most three nonlinear monolithic characters. We give some sufficient conditions for the inequality related to monolithic characters or real-valued irreducible characters of $G$ when the commutator subgroup of $G$ is supersolvable.

We prove that if $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$ is a space satisfying the discrete countable chain condition with a rank 3-diagonal then the cardinality of $X$ is at most $\mathfrak{c}$.

Let G be a group. A subset X of G is a set of pairwise noncommuting elements if xy≠yx for any two distinct elements x and y in X. If |X|≥|Y | for any other set of pairwise noncommuting elements Y in G, then X is said to be a maximal subset of pairwise noncommuting elements. In this paper we determine the cardinality of a maximal subset of pairwise noncommuting elements for some p-groups of maximal class. Specifically, we determine this cardinality for all 2 -groups and 3 -groups of maximal class.

For a prime powerq, let 𝔽qbe the finite field ofqelements. We show that 𝔽*q⊆d𝒜2for almost every subset 𝒜⊂𝔽qof cardinality ∣𝒜∣≫q1/d. Furthermore, ifq=pis a prime, and 𝒜⊆𝔽pof cardinality ∣𝒜∣≫p1/2(logp)1/d, thend𝒜2contains both large and small residues. We also obtain some results of this type for the Erdős distance problem over finite fields.

Let G be a group. A subset X of G is said to be nonnilpotent if for any two distinct elements x and y in X, 〈x,y〉 is a nonnilpotent subgroup of G. If, for any other nonnilpotent subset X′ in G, ∣X∣≥∣X′ ∣, then X is said to be a maximal nonnilpotent subset and the cardinality of this subset is denoted by ω(𝒩G) . Using nilpotent nilpotentizers we find a lower bound for the cardinality of a maximal nonnilpotent subset of a finite group and apply this to the general linear group GL (n,q) . For all prime powers q we determine the cardinality of a maximal nonnilpotent subset of the projective special linear group PSL (2,q) , and we characterize all nonabelian finite simple groups G with ω(𝒩G)≤57 .

For three points$\vec {u}$,$\vec {v}$and$\vec {w}$in then-dimensional space 𝔽nqover the finite field 𝔽qofqelements we give a natural interpretation of an acute angle triangle defined by these points. We obtain an upper bound on the size of a set 𝒵 such that all triples of distinct points$\vec {u}, \vec {v}, \vec {w} \in \cZ $define acute angle triangles. A similar question in the real space ℛndates back to P. Erdős and has been studied by several authors.

Let G be a group. A subset N of G is a set of pairwise noncommuting elements if xy⁄=yx for any two distinct elements x and y in N. If ∣N∣≥∣M∣ for any other set of pairwise noncommuting elements M in G, then N is said to be a maximal subset of pairwise noncommuting elements. In this paper we determine the cardinality of a maximal subset of pairwise noncommuting elements in a three-dimensional general linear group. Moreover, we show how to modify a given maximal subset of pairwise noncommuting elements into another maximal subset of pairwise noncommuting elements that contains a given ‘generating element’ from each maximal torus.

A special case of the main result is the following. LetGbe a finite, non-supersoluble group in which from arbitrary subsetsX,Yof cardinalitynwe can always findx∈Xandy∈Ygenerating a supersoluble subgroup. Then the order ofGis bounded by a function ofn. This result is a finite version of one line of development of B.H. Neumann's well-known and much generalised result of 1976 on infinite groups.

Given two different positive integers k and l, a (k, l)-free set of some group (G, +) is defined as a set  ⊂ G such that k∩l = ∅. This paper is devoted to the complete determination of the structure of (k, l)-free sets of ℤ/pℤ (p an odd prime) with maximal cardinality. Except in the case where k = 2 and l = 1 (the so-called sum-free sets), these maximal sets are shown to be arithmetic progressions. This answers affirmatively a conjecture by Bier and Chin which appeared in a recent issue of this Bulletin.

Let n be a positive integer or infinity (denoted ∞), k a positive integer. We denote by Ωk(n) the class of groups G such that, for every subset X of G of cardinality n + 1, there exist distinct elements x, y ∈ X and integers t0, t1…, tk such that , where xi, ∈ {x, y}, i = 0, 1,…,k, x0 ≠ x1. If the integers t0, t1,…,tk are the same for any subset X of G, we say that G is in the class Ω̅k(n). The class k (n) is defined exactly as Ωk(n) with the additional conditions . Let t2, t3,…,tk be fixed integers. We denote by the class of all groups G such that for any infinite subsets X and Y there exist x ∈ X, y ∈ Y such that , where xi ∈ {x, y}, x0 ≠ x1, i = 2, 3, …, k. Here we prove that (1) If G ∈ k(2) is a finitely generated soluble group, then G is nilpotent.(2) If G ∈ Ωk(∞) is a finitely generated soluble group, then G is nilpotentby-finite.(3) If G ∈ Ω̅k(n), n a positive integer, is a finitely generated residually finite group, then G is nilpotent-by-finite.(4) If G is an infinite -group in which every nontrivial finitely generated subgroup has a nontrivial finite quotient, then G is nilpotent-by-finite.

Quasi reflexive Banach spaces are characterised among the weakly countably determined Asplund spaces, in terms of the cardinality of the sets of linearly independent bounded linear functionals each of which does not attain its supremum on the unit sphere.

For an associative ring R, we investigate the relation between the cardinality of the commutator [R, R], or of higher commutators such as [[R, R], [R, R]], the cardinality of the ideal it generates, and the index of the centre of R. For example, when R is a semiprime ring, any finite higher commutator generates a finite ideal, and if R is also 2-torsion free then there is a central ideal of R of finite index in R. With the same assumption on R, any infinite higher commutator T generates an ideal of cardinality at most 2cardT and there is a central ideal of R of index at most 2cardT in R.

Given a finite set S of cardinality N, the minimum number of j-subsets of S needed to cover all the r-subsets of S is called the covering number C(N, j, r). While Erdös and Hanani's conjecture that was proved by Rödl, no nontrivial upper bound for C(N, j, r) was known for finite N. In this note we obtain a nontrivial upper bound by showing that for finite N

A critical set in a Latin square of order n is a set of entries in an n×n array which can be embedded in precisely one Latin square of order n, with the property that if any entry of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order n. The number of critical sets grows super-exponentially as the order of the Latin square increases. It is difficult to find patterns in Latin squares of small order (order 5 or less) which can be generalised in the process of creating new theorems. Thus, I have written many algorithms to find critical sets with various properties in Latin squares of order greater than 5, and to deal with other related structures. Some algorithms used in the body of the thesis are presented in Chapter 3; results which arise from the computational studies and observations of the patterns and subsequent results are presented in Chapters 4, 5, 6, 7 and 8. The cardinality of the largest critical set in any Latin square of order n is denoted by lcs(n). In 1978 Curran and van Rees proved that lcs(n) =2) and m×2^α+1 (m odd, α>=2 and α≠3), and a new lower bound on lcs(4m). It also discusses critical sets in intercalate-rich Latin squares of orders 11 and 14. In Chapter 6 a construction is given which verifies the existence of a critical set of size n²÷ 4 + 1 when n is even and n>=6. The construction is based on the discovery of a critical set of size 17 for a Latin square of order 8. In Chapter 7 the representation of Steiner trades of volume less than or equal to nine is examined. Computational results are used to identify those trades for which the associated partial Latin square can be decomposed into six disjoint Latin interchanges. Chapter 8 focusses on critical sets in Latin squares of order at most six and extensive computational routines are used to identify all the critical sets of different sizes in these Latin squares.

The main concern of this thesis is the study of problems involving collections of subsets of a finite set [n] = {1,2,...,n} and especially antichains on [n]. An antichain is a collection of sets in which no two sets are comparable under set inclusion. An antichain A is flat if there exists an integer k ≥ 0 such that every set in A has cardinality either k or k + 1. The size of A is |A| and the volume of A is ∑A∈A |A|. A unifying problem in the thesis is the flat antichain conjecture which states that A is an antichain on [n] if and only if there exists a flat antichain on [n] with the same size and volume as A. The truth of the conjecture would provide a simple test for the existence of antichains on [n] with a given size and volume. The flat antichain conjecture is known to hold in several special cases. This thesis shows that it holds in several further cases. Two main approaches have been taken in the investigation of the flat antichain conjecture. The first approach consists of studying the volumes of antichains. Building on earlier work of Clements we observe that for given n and s, the antichains on [n] of size s which achieve minimum (maximum) volume are flat antichains, where the minimum (maximum) is taken over all antichains on [n] of size s. Further, we show that if A is an antichain on [n] then there exists a flat antichain on [n] with the same volume as A. In proving this last result we prove that the flat antichain conjecture holds in one special case. The second approach involves counting the number of subsets and supersets of certain collections of sets as defined below. The squashed (or colex) order on sets is a set ordering with the property that the number of subsets of a collection of k-sets is minimised when the collection consists of an initial segment of k-sets in squashed order. In the universal set [n], consider the collections Lk+1(p) and Lk-1(p) consisting of the last p (k + 1)-sets in squashed order and the last p (k - 1)-sets in squashed order respectively. Let a1 be the number of supersets of size k of the sets in Lk-1(p). Let a2 be the number of subsets of size k of the sets in Lk+1(p) which are not subsets of any (k + 1)-set preceding the sets in Lk+1(p) in squashed order. We show that a1 + a2 > 2p. This result, called the 3-levels result, enables us to show that the flat antichain conjecture holds in several cases. Several conjectures and open problems which are related to the 3-levels result are stated and implications of the truth of the flat antichain conjecture are considered.

A Grothendieck topos has the property that its Yoneda embedding has a left-exact left adjoint. A category with the latter property is called lex-total. It is proved here that every lex-total category is equivalent to its category of canonical sheaves. An unpublished proof due to Peter Freyd is extended slightly to yield that a lex-total category, which has a set of objects of cardinality at most that of the universe such that each object in the category is a quotient of an object from that set, is necessarily a Grothendieck topos.

The following theorem is proved.Let X be a finite set of cardinality n ≥ 2, and let F be a family of subsets of X. Suppose that for F1, F2, F3 ∈ F we have |F1 ∩ F2 ∩ F3| ≥ 2. Then |F| ≤ 2n−2with equality holding if and only if for two different elements x, y of X, F = {F ⊆ X | x ∈ F, y ∈ F}.

Let F be a family of k-element subsets of an n-set, n > n0(k). Suppose any two members of F have non-empty intersection. Let τ(F) denote min|T|, T meets every member of F. Erdös, Ko and Rado proved and that if equality holds then τ(F) = 1. Hilton and Milner determined max|F| for τ(F) = 2. In this paper we solve the problem for τ(F) = 3.The extremal families look quite complicated which shows the power of the methods used for their determination.

The degree of disjunction, δ(F), of a family F of functions is the least cardinal τ such that every pair of functions in F agree on a set of cardinality less than τ.Suppose θ, μ, λ, κ are non-zero cardinals with θ ≤ μ ≤ λ. This paper is concerned with functions which map μ-sized subsets of λ into κ. We first show there is always a ‘large’ family F of such functions satisfying δ(F) ≤ θ. Next we determine the cardinalities of families F of such functions that are maximal with respect to δ(F) ≤ θ.

Let X be a finite set of cardinality n, and let F be a family of k-subsets of X. In this paper we prove the following conjecture of P. Erdös and V.T. Sós.If n > n0(k), k ≥ 4, then we can find two members F and G in F such that |F ∩ G| = 1.

It is known from [2] that every generalised ordered (GO) space is paracompact if and only if it has no closed subspace which is homeomorphic to a stationary set in some regular uncountable cardinal (for more details, see [4, 3]). In view of this result, I conjectured that for every regular uncountable cardinal K, every GO space is «;-paracompact if and only if it has no closed subspace which is homeomorphic to a stationary set in K. But unfortunately, I found a counterexample at once. The ordered space u>! is not u>2-paracompact, but it has no closed subspace which is homeomorphic to a stationary set in w2, since the cardinality of u>i is less than that of u>2 . Recently, I proved in [5] that every GO space is paracompact if and only if it has the S-property (defined below) using the result of [2], also that every GO space has the shrinking property (defined below). In this paper, we shall show that for every uncountable regular cardinal K, every GO space has the B(»c)-property if and only if it has no closed subspace which is homeomorphic to a stationary set in K. Furthermore, we shall also clarify the relation between /c-paracompactness and the S(/t)-property in GO spaces.

A new notion of α-connectedness (α-path connectedness) in general topological spaces is introduced and it is proved that for a real-valued function defined on a space with this property, the cardinality of the antipodal coincidence set is at least as large as the cardinal number α. In particular, in linear topological spaces, the antipodal coincidence set of a real-valued function has cardinality at least that of the continuum. This could be regarded as a treatment of some Borsuk-Ulam type results in the setting of general topology.

The thesis is concerned with a number of problems in Combinatorial Set Theory. The Generalized Continuum Hypothesis is assumed. Suppose X and K are non-zero cardinals. By successively identifying K with airwise disjoint sets of power K, a function/: X-*•K can be viewed as a transversal of a pairwise disjoint (X, K)family A . Questions about families of functions in K can thus bethought of as referring to families of transversals of A. We wish to consider generalizations of such questions to almost disjoint families; in particular we are interested in extensions of the following two problems: (i) What is the 'maximum' cardinality of an almost disjoint family of functions each mapping X into K? (ii) Describe the cardinalities of maximal almost disjoint families of functions each mapping X into K. Article in Bulletin of the Australian Mathematical Society 27(03):477 - 479 · June 1983

Theorem 1 and Corollary 1 of the original paper [Bull. Austral. Math. Soc. 25 (1982), 99–101] are false unless, for example, we assume the Generalised Continuum Hypothesis. The proof of Theorem 1 correctly shows that exp(|c|) ≤ exp(|D|) but one cannot then deduce that the cardinality of C is at most that of D. All one can deduce is that the cardinality of C is less than exp(|D|).