The pointwise order makes the group C(X) of continuous real-valued functions on a topological space X a lattice-ordered group. We give a characterization of the compatible tight Riesz orders on C(X), and also of their maximal tangents, in terms of the zero-sets of X. The space of maximal tangents of a given compatible tight Riesz order T is studied, and consequently the concept of the T-radical of C(X) is introduced, the T-radical being the intersection of all the maximal tangents of T.

All such nonabelian finite p-groups are classified. They coincide with the class of nonabelian finite p-subgroups of GL(p, F), where F is a field, not of characteristic p, which contains all p power roots of 1, or agian with the class of all nonabelian finite subgroups of Zp= wr Zp. Various automorphism groups associated to them and their representations are calculated. Two such subgroups of GL(p, F) are conjugate as subgroups of GL(p, F) iff they are isomorphic.

(Point, closed subset)-separation axioms and closed subsets separation axioms for topological spaces will be uniformly defined. Then it is shown that a subcategory of TOP is bireflective in TOP if and only if Ob consists of all separated spaces for some (point, closed subset)-separation axiom. A characterization theorem on subcategories of all separated spaces for closed subsets separation axioms is also given by using the category SEP of all separation spaces and the embedding functor G: TOP → SEP. As an application we have that a T1-space is normal if and only if it is embedded in a product space of the unit intervals in SEP.

Any congruence on an orthodox semigroup S induces a partition of the set E of idempotents of S satisfying certain normality conditions. Meakin (1970) has characterized those partitions of E which are induced by congruences on S as well as the largest congruence ρ and the smallest congruence σ on S corresponding to such a partition of E. In this paper a more precise description of ρ and σ is given.

This paper gives theorems which encompass known characterizations of many of the generalized compactness properties.

My attention has been called to the fact that Theorem E1, quoted on p. 445 of my paper “On the embedding of a group in a join of given groups’, J. Austral. Math. Soc. 17 (1974), 434–495, is practically identical with Théoréme I (p. 47) of the second part of the three-part paper “Produit complet des groupes de permutations et probléme d'extension des groupes” by M. Krasner and L. Kaloujnine, Acta Sci. Math. Szeged 14 (1951), 39–66. My oversight in referring only to the third part of this important work, which contains the special case often quoted as the Kaloujnine-Krasner Theorem, is regrettable: it is the much more general and definitive Théoréme I which should be equally or better known by that name.

A class of algebras A is said to have the strong amalgamation property if for any indexed set of algebras {Ai: i ∈ J} from A, each having an algebra U ∈ A as a subalgebra, there exist an algebra B ∈ A and monomorphisms φi: Ai → B (for each i ∈ J) such that (i) φi|U = φi| U for all i,j∈J, (ii) φi(Ai) ∩ φi(Aj) =φi(U) for atl i, j ∈ J with i ≠ j, where φi|U denotes the restriction of φi to U. Omitting condition (ii) gives us the definition of the weak amalgamation property.

Let L = HK be a semidirect product of a normal locally finite π′-group H by a locally finite π′-group K, where π, is a set of primes. Suppose CK(H) = 1 and L is Sylow π-sparse (which in the countable case just says that the Sylow π-subgroups of L are conjugate). This paper completes the characterization of those groups which can occur as K—this had previously been obtained under the assumption that L is locally soluble. The answer is the same—essentially that the groups occurring are those having a subgroup of finite index which is a subdirect product of so-called “pinched” groups.

An (r, 1) system is pair (V, F) where V is a v-set and F is a family of non-null subsets of V (b in number) which satisfy the following.

(i) Every pair of distinct members of V occur in precisely one member of F;

(ii) Every member of V occurs in precisely r members of F.

A pseudo parallel complement PPC(n, α) is an (n+1, 1) system with ν = n2 – αn and b ≦ n2 + n – α in which there are at least n – α blocks of size n. A pseudo intersection complement PIC (n, α) is an (n+1, 1) system with ν=n2 – αn + α – 1 b ≦ n2+n – α in which there are at least n – α+1 blocks of size n-1. It is shown that for α ≧ 4, every PIC(n, α) can be embedded in a PPC(n, α – 1) and that for n > (α4 + 2α3 + 2α2 + α)/2 every PPC(n, α) can be embedded in a projective plant of order n. The latter generalizes a result of Totten (who proves the result for α = 1).

Conditions are investigated under which a product G = HK of soluble subgroups H, K, each permutable in G, is soluble. It is shown in particular that this is true if one of the subgroups is nilpotent or if H ∩ K is subnormal in both H and K.

Let G be a finitely generated metabelian group whose derived group G′ has finite rank. It is shown that G can be embedded in a finitely presented metabelian group H with H′ of finite rank.