page_title: Artin transfer (group theory)
section_title: TTT singulets
hierarchical_section_title: Artin transfer (group theory) / Inheritance from quotients / TTT singulets
caption_attribution_description: English: Artin transfer - Figure 3: Epimorphisms and derived quotients. User:DanielConstantinMayer added png-file "Epimorphisms and Artin transfers"
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: Proposition. Assume  are as above and  is the image of a subgroup  The commutator subgroup of  is the image of the commutator subgroup of  Therefore,  induces a unique epimorphism , and thus  is a quotient of  Moreover, if , then the map  is an isomorphism (See Figure 2).
  Proof. This claim is a consequence of the Main Theorem in the article on the induced homomorphism. Nevertheless, an independent proof is given as follows: first, the image of the commutator subgroup is 
  Second, the epimorphism  can be restricted to an epimorphism . According to the previous section, the composite epimorphism  factors through  by means of a uniquely determined epimorphism  such that . Consequently, we have . Furthermore, the kernel of  is given explicitly by .
  Finally, if , then  is an isomorphism, since .
  Definition. Due to the results in the present section, it makes sense to define a partial order on the set of abelian type invariants by putting , when , and , when .
caption: Figure 3: Epimorphisms and Artin transfers.
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