Abstract
Based on a general theory of descendant trees of finite p-groups and the virtual periodicity isomorphisms between the branches of a coclass subtree, the behavior of algebraic invariants of the tree vertices and their automorphism groups under these isomorphisms is described with simple transformation laws. For the tree of finite 3-groups with elementary bicyclic commutator quotient, the information content of each coclass subtree with metabelian mainline is shown to be finite. As a striking novelty in this paper, evidence is provided of co-periodicity isomorphisms between coclass forests which reduce the information content of the entire metabelian skeleton and a significant part of non-metabelian vertices to a finite amount of data.
Highlights
Denote by the rooted tree of all finite 3-groups G with elementary bicyclic commutator quotient G G′ C3 × C3, and let ∗ be the infinite pruned subtree of, where all descendants of capable non-metabelian vertices are eliminated
Based on a general theory of descendant trees of finite p-groups and the virtual periodicity isomorphisms between the branches of a coclass subtree, the behavior of algebraic invariants of the tree vertices and their automorphism groups under these isomorphisms is described with simple transformation laws
As a striking novelty in this paper, evidence is provided of co-periodicity isomorphisms between coclass forests which reduce the information content of the entire metabelian skeleton and a significant part of non-metabelian vertices to a finite amount of data
Summary
Denote by the rooted tree of all finite 3-groups G with elementary bicyclic commutator quotient G G′ C3 × C3 , and let ∗ be the infinite pruned subtree of , where all descendants of capable non-metabelian vertices are eliminated. Between the infinite coclass forests (r ) , r ≥ 1 , which reduce the information content of the pruned tree ∗ to the union of pre-period ( ) ( ) (r) 1≤r≤4 and first primitive period (r ) 5≤r≤6 , consisting of the leading six coclass forests only. The discovery of this co-periodicity is the progressive innovation in the present paper. 1 ≤ j ≤ t , of the coclass forests (r ) , and ([6], §5.2, pp. 114-116) establishes the connection between the coclass trees r j and infinite metabelian pro-3 groups of coclass r
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