Space groups and groups of prime-power order I

Periodic patterns in the graph of p-groups of maximal class

“…The first major results are due to Blackburn [2] who also obtained a full classification of the 2-and 3-groups of maximal class. Motivated by Blackburn's success, the pgroups of maximal class became a well-studied type of p-groups and, as a generalization, Leedham-Green and Newman [15] defined the coclass of a group of order p n and nilpotency class c as n À c.…”

Profinite Groups with Restricted Centralizers

“…Example 3. Let G be a /»-adic space group with a cyclic point group; that is, G is an extension of a finitely generated free /?-adic module by a cyclic /?-group acting faithfully and irreducibly (see [5] for background and basic properties). Then G is a pro-/?…”

Gradings of non-graded Hamiltonian Lie algebras

“…In particular, having finite coclass for a pro-p-group G means that all lower central 400 A. Caranti and S. Mattarei [2] quotients yi(G)/ f y i+ \(G) have order at most p from some point on. The five coclass conjectures advanced in [45] are now theorems thanks to the efforts of several authors, culminating in [42] and [51]. They give information on pro-p-groups of finite coclass (Conjecture C, the simplest to state, claiming that every pro-p-group of finite coclass is soluble), but also asymptotic information on families of finite p-groups of fixed coclass.…”

“…This trend was initiated by Leedham-Green and Newman in 1980, who proposed in [45] one way of getting around the universally believed impossibility of a classification of p-groups up to isomorphism. One of their intuitions was that of using the coclass rather than the (nilpotency) class of p-group as a fundamental invariant.…”