Countable and Uncountable in Group Theory

A prominent, recurring feature of group theory has been the determination of groups (all of) whose subgroups possess some group theoretical property. For infinite groups in a suitable universe a number of different approaches have been used in this regard. For locally finite groups, for example, knowledge of the structure of the finite subgroups is often crucial. On the other hand the concept of "largeness" has also recently played an interesting role. Moving from this, I started to study how subgroups of uncountable cardinality affect an uncountable group. Let X be a group theoretical proper, let G be a group of uncountable cardinality and suppose that all its proper uncountable subgroups satisfy X. Is it true that all (proper) subgroups of G satisfy X? The thesis exploits this question, showing that, under some soluble conditions, the answer is often positive. Finally the thesis deals with countably recognizable properties, which has a strong relation with the previous question.