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 (with a prime number
p), 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. These strategies of
pattern recognition are useful in purely group theoretic context, as well as for applications in
algebraic number theory concerning Galois groups of higher
p-class fields and Hilbert
p-class field towers.