On the classification of prime-power groups by coclass

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

“…A similar theorem has been proved in a more general context by Eick and Leedham-Green [7]. Theorem 9.1 strongly supports the conjecture that the p-groups of maximal class can be classified by finitely many parametrized presentations.…”

“…Du Sautoy [6] and Eick and Leedham-Green [7] proved a periodic pattern in coclass trees for arbitrary prime and coclass. Eick and Leedham-Green also provide explicit bounds for the parameters of this periodicity.…”

“…Eick and Leedham-Green [7] carried out a detailed structure analysis of the groups in a coclass graph. Based on this, they defined periodicity classes of groups which underpin the graph isomorphisms.…”

The distribution of second p-class groups on coclass graphs

“…We adopt the most recent view of coclass graphs, which is given by Eick and LeedhamGreen [33], and by Dietrich, Eick, Feichtenschlager [27, p. 46]. …”

“…Periodic patterns. An important purpose of this paper is to emphasize that coclass graphs G(p, r) are particularly well suited for visualizing periodic properties [30,33] of infinite sequences of finite p-groups G, such as parametrized power-commutator presentations [15,66], automorphism groups Aut(G), Schur multipliers H 2 (G, Z p ) and other cohomology groups of G, transfer kernel types κ(G) [59], transfer target types τ (G) [60], and defect of commutativity k(G) expressed by the depth dp(G) (Corollaries 3.1.1 and 3.10.1). In number theoretic applications, selection rules for second p-class groups G = G 2 p (K) of special base fields K [58] are additional periodic properties.…”