Hall's universal group

Hall's universal group, often denoted as \( H \), is a type of infinite group that arises in group theory, specifically in the context of group actions and representations. It is named after Philip Hall, who introduced it in the context of group theory. More specifically, Hall's universal group can be thought of as the group of finitely generated groups or, in a broader sense, the group of groups that allows one to categorize all groups that satisfy certain properties.