page_title: Artin transfer (group theory)
section_title: Abelianization of type (p,p)
hierarchical_section_title: Artin transfer (group theory) / Systematic library of SDTs / Coclass 2 / Abelianization of type (p,p)
caption_attribution_description: English: This diagram shows the descendant tree of the root &lt;243,8&gt;, restricted to finite 3-groups with coclass 2. All vertices possess an abelianization of type (3,3). Important vertices are labelled with their SmallGroups Library identifier in angle brackets. TTT means transfer target type. TKT means transfer kernel type. The mainline descendant &lt;729,54&gt; of the root was called U by Judith A. Ascione.
context_page_description: In the mathematical field of group theory, an Artin transfer is a certain homomorphism from an arbitrary finite or infinite group to the commutator quotient group of a subgroup of finite index. Originally, such mappings arose as group theoretic counterparts of class extension homomorphisms of abelian extensions of algebraic number fields by applying Artin's reciprocity maps to ideal class groups and analyzing the resulting homomorphisms between quotients of Galois groups. However, independently of number theoretic applications, a partial order on the kernels and targets of Artin transfers has recently turned out to be compatible with parent-descendant relations between finite p-groups, which can be visualized in descendant trees. Therefore, Artin transfers provide a valuable tool for the classification of finite p-groups and for searching and identifying particular groups in descendant trees by looking for patterns defined by the kernels and targets of Artin transfers.
context_section_description: Three coclass trees, ,  and  for , are endowed with information concerning TTTs and TKTs.
  On the tree , the -groups of coclass  with bicyclic centre in Figure 8 can be defined by the following parametrized polycyclic pc-presentation.
  5
  where the nilpotency class is , the order is  with , and  are parameters.
  The branches are strictly periodic with pre-period  and period length , and have depth  and width .
  Polarization occurs for the first component and the TTT is , only dependent on .
  The TKT depends on the parameters and is
   for the mainline vertices with ,
   for the capable vertices with ,
   for the terminal vertices with ,
  and  for the terminal vertices with .
  Mainline vertices and vertices on even branches are -groups.
  On the tree , the -groups of coclass  with bicyclic centre in Figure 9 can be defined by the following parametrized polycyclic pc-presentation.
  6
  where the nilpotency class is , the order is  with , and  are parameters.
  The branches are strictly periodic with pre-period  and period length , and have depth  and width .
  Polarization occurs for the second component and the TTT is , only dependent on .
  The TKT depends on the parameters and is
   for the mainline vertices with ,
   for the capable vertices with ,
   for the terminal vertices with ,
  and  for the terminal vertices with .
  Mainline vertices and vertices on even branches are -groups.
caption: Figure 9: Second structured descendant tree of 3-groups with coclass 2 and abelianization (3,3).
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