page_title: Artin transfer (group theory)
section_title: Connections between layers
hierarchical_section_title: Artin transfer (group theory) / Abelianization of type (p2,p) / Connections between layers
caption_attribution_description: English: This commutative diagram of group homomorphisms shows the map G/G' -&gt; A on the derived quotient of G, induced by a homomorphism from a group G into an abelian group A, where G -&gt; G/G' denotes the natural projection. User:DanielConstantinMayer added the png-file "Factorizing through the derived quotient"
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: The Artin transfer  is the composition  of the induced transfer  from  to  and the Artin transfer  
  There are two options regarding the intermediate subgroups
  For the subgroups  only the distinguished maximal subgroup  is an intermediate subgroup.
  For the Frattini subgroup  all maximal subgroups  are intermediate subgroups.
  This causes restrictions for the transfer kernel type  of the second layer, since
  and thus
  But even
  Furthermore, when  with  an element  of order  with respect to , can belong to  only if its th power is contained in , for all intermediate subgroups , and thus: , for certain , enforces the first layer TKT singulet , but , for some , even specifies the complete first layer TKT multiplet , that is , for all .
caption: Figure 1: Factoring through the abelianization.
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