This thesis is in two parts. The first is a study of soluble profinite groups with the maximal condition (max) and mirrors the development of the fundamental properties of polycyclic groups. Two of the main results are a profinite version of Hall's criterion for nilpotence and the existence of nilpotent almost-supplements for the Fitting subgroup. It is shown that the automorphism group of a soluble profinite group with max itself has max and that a soluble profinite group has max if and only if each of its subnormal subgroups of defect at most 2 has max. Quantitative versions of these results are also obtained. The second part derives a new characterization of abstract and profinite branch groups. This is achieved by examining the subnormal subgroup structure of just non-(virtually abelian) groups and a partial classification of this larger class is obtained. Weak branch groups are shown to satisfy no abstract group laws and to contain many abstract free subgroups in the profinite case. The notion of a branch group is weakened to that of a generalized branch group. Associated with each generalized branch group is its structure graph, and the circumstances under which this graph is a tree are characterized.

This thesis studies some classes of non-periodic branch groups. In particular their growth, relations between elements and their Hausdorff dimensions.

This thesis is a study of the subgroup structure of some remarkable groups of automorphisms of rooted trees. It is divided into two parts. The main result of the first part is seemingly of an algorithmic nature, establishing that the Gupta--Sidki 3-group G has solvable membership problem. This follows the approach of Grigorchuk and Wilson who showed the same result for the Grigorchuk group. The proof, however, is not algorithmic, and it moreover shows a striking subgroup property of G: that all its infinite finitely generated subgroups are abstractly commensurable with either G or G × G. This is then used to show that G is subgroup separable which, together with some nice presentability properties of G, implies that the membership problem is solvable. The proof of the main theorem is also used to show that G satisfies a "strong fractal" property, in that every infinite finitely generated subgroup acts like G on some rooted subtree. The second part concerns the subgroup structure of branch and weakly branch groups in general. Motivated by a natural question raised in the first part, a necessary condition for direct products of branch groups to be abstractly commensurable is obtained. From this condition it follows that the Gupta--Sidki 3-group is not abstractly commensurable with its direct square. The first main result in the second part states that any (weakly) branch action of a group on a rooted tree is determined by the subgroup structure of the group. This is then applied to answer a question of Bartholdi, Siegenthaler and Zalesskii, showing that the congruence subgroup property for branch and weakly branch groups is independent of the actions on a tree. Finally, the information obtained on subgroups of branch groups is used to examine which groups have an essentially unique branch action and why this holds.