On the Method of Direct Products in the Theory of Quasi-Invariant Measures

For invariant (quasi-invariant) σ-finite measures on an uncountable group (𝐺, ·), the behaviour of small sets with respect to the operation “·” is studied. Some classes of non-commutative groups (𝐺, ·) are discussed especially, by using a representation of the original group in the form of a direct product of its two subgroups, one of which is commutative.

Extending Quasi-Invariant Measures by Using Subgroups of a Given Group

A method of extending σ-finite quasi-invariant measures given on an uncountable group, by using a certain family of its subgroups, is investigated.

Embedding into groups with well-described lattices of subgroups

A thrifty embedding scheme of an arbitrary set of groups in a simple infinite group with a given outer automorphism group is presented. One of the applications of this scheme is the existence (assuming CH) of an uncountable group G in which all proper subgroups are countable such that G contains every countable group.

On the number of normal subgroups of an uncountable group

In this paper two theorems are proved that give a partial answer to a question posed by G. Behrendt and P. Neumann. Firstly, the existence of a group of cardinality ℵ1 with exactly ℵ1 normal subgroups, yet having a subgroup of index 2 with 2ℵ1 normal subgroups, is consistent with ZFC (the Zermelo-Fraenkel axioms for set theory together with the Axiom of Choice). Secondly, the statement “Every metabelian-by-finite group of cardinality ℵ1 has 2ℵ1 normal subgroups” is consistent with ZFC.

Ext(Q,Z) is the additive group of real numbers

Standard homological methods and a theorem of Harrison on cotorsion groups are used to prove the result mentioned.In this note Z denotes an infinite cyclic group, Q the additive group of rational numbers, Zp ∞ a p–quasicyclie group, and Ip the group of p–adic integers.Pascual Llorente proves in [3] that Ext(Q,z) is an uncountable group, and gives explicitly a countably infinite subset. Very little extra effort produces the result embodied in the title, as follows.

Some Sylow subgroups

Most of the well-known theorems of Sylow for finite groups and of P. Hall for finite soluble groups have been extended to certain restricted classes of infinite groups. To show the limitations of such generalizations, examples are here constructed of infinite groups subject to stringent but natural restrictions, groups in which certain Sylow or Hall theorems fail. All the groups are metabelian and of exponent 6. There are countable such groups in which a Sylow 2-subgroup has a complement but no Sylow 3-complement; or again no complement at all. There are countable such groups with continuously many mutually non-isomorphic Sylow 3-subgroups. There are groups, necessarily of uncountable order, with Sylow 2-subgroups of different orders. The most elaborate example is of an uncountable group in which all Sylow 2-subgroups and 3-subgroups are countable, and none is complemented.