page_title: Artin transfer (group theory)
section_title: Abelianization of type (p,p,p)
hierarchical_section_title: Artin transfer (group theory) / Systematic library of SDTs / Coclass 3 / Abelianization of type (p,p,p)
caption_attribution_description: English: This diagram shows the descendant tree of the root &lt;243,6&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. Vertices surrounded by ovals are populated by second 3-class groups of complex quadratic fields with discriminant d&lt;0. The underlined counters adjacent to the ovals surrounding these vertices give the minimal absolute discriminants of the distribution. TTT means transfer target type. TKT means transfer kernel type. The mainline descendant &lt;729,49&gt; of the root was called Q by Judith A. Ascione. User:DanielConstantinMayer added png-file "MinDiscriminantsTreeQ"
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: and  for ,
   and  for .
caption: Figure 10: Minimal discriminants for the first ASCT of 3-groups with coclass 2 and abelianization (3,3).
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