GROUPS WITH MANY PRONORMAL SUBGROUPS

A subgroup H of a group G is pronormal in G if each of its conjugates $H^g$ in G is conjugate to it in the subgroup $\langle H,H^g\rangle $ ; a group is prohamiltonian if all of its nonabelian subgroups are pronormal. The aim of the paper is to show that a locally soluble group of (regular) cardinality in which all proper uncountable subgroups are prohamiltonian is prohamiltonian. In order to obtain this result, it is proved that the class of prohamiltonian groups is detectable from the behaviour of countable subgroups. Examples are exhibited to show that there are uncountable prohamiltonian groups that do not behave very well. Finally, it is shown that prohamiltonicity can sometimes be detected through the analysis of the finite homomorphic images of a group.

Uncountable groups in which normality is a transitive relation

A group is called a [Formula: see text]-group if all its subnormal subgroups are normal. It is proved here that if [Formula: see text] is an uncountable soluble periodic group of regular cardinality [Formula: see text] in which all subnormal subgroups of cardinality [Formula: see text] are normal, then [Formula: see text] is a [Formula: see text]-group, provided that it satisfies a suitable residual condition. An example shows that this latter restriction cannot be avoided.

On cardinalities and compact closures

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>We show that there exists a Hausdorff topology on the set </span><span>R </span><span>of real numbers such that a subset </span><span>A </span><span>of </span><span>R </span><span>has compact closure if and only if </span><span>A </span><span>is countable. More generally, given any set </span><span>X </span><span>and any infinite set </span><span>S</span><span>, we prove that there exists a Hausdorff topology on </span><span>X </span><span>such that a subset </span><span>A </span><span>of </span><span>X </span><span>has compact closure if and only if the cardinality of </span><span>A </span><span>is less than or equal to that of </span><span>S</span><span>. When we attempt to replace “than than or equal to” in the preceding statement with “strictly less than,” the situation is more delicate; we show that the theorem extends to this case when </span><span>S </span><span>has regular cardinality but can fail when it does not. This counterexample shows that not every bornology is a bornology of compact closure. These results lie in the intersection of analysis, general topology, and set theory. </span></p></div></div></div>

A nilpotent-generated semigroup associated with a semigroup of full transformations

SynopsisLet X be a set with infinite regular cardinality m and let ℱ(X) be the semigroup of all self-maps of X. The semigroup Qm of ‘balanced’ elements of ℱ(X) plays an important role in the study by Howie [3,5,6] of idempotent-generated subsemigroups of ℱ(X), as does the subset Sm of ‘stable’ elements, which is a subsemigroup of Qm if and only if m is a regular cardinal. The principal factor Pm of Qm, corresponding to the maximum ℱ-class Jm, contains Sm and has been shown in [7] to have a number of interesting properties.Let N2 be the set of all nilpotent elements of index 2 in Pm. Then the subsemigroup (N2) of Pm generated by N2 consists exactly of the elements in Pm/Sm. Moreover Pm/Sm has 2-nilpotent-depth 3, in the sense that

A class of bisimple, idempotent-generated congruence-free semigroups

SynopsisLet X be a set with infinite regular cardinality m. Within the full transformation semigroup ℑ(X) a subsemigroup Sm is described which is bisimple and idempotent-generated. Its minimum non-trivial homomorphic image has both these properties and is also congruence-free. The semigroup contains an isomorphic copy of every semigroup having order less than m.