View source for Transfer (group theory)

In the mathematical field of [[Group theory|group theory]], the '''transfer''' defines, given a [[Group (mathematics)|group]] ''G'' and a [[Subgroup|subgroup]] ''H'' of finite [[Index of a subgroup|index]], a [[Group homomorphism|group homomorphism]] from ''G'' to the abelianization of ''H''. It can be used in conjunction with the [[Sylow theorems]] to obtain certain numerical results on the existence of finite simple groups.
The transfer was defined by {{harvs|txt|first=Issai|last= Schur|year=1902}} and rediscovered by {{harvs|txt|first=Emil|last= Artin|authorlink=Emil Artin|year=1929}}.<ref name=S122/>

==Construction==
The construction of the map proceeds as follows:<ref>Following Scott 3.5</ref> Let [''G'':''H''] = ''n'' and select [[Coset|coset]] [[Artin transfer (group theory)#Transversals of a subgroup|representatives]], say 
:<math>x_1, \dots, x_n,\,</math>
for ''H'' in ''G'', so ''G'' can be written as a disjoint union
:<math>G = \bigcup\ x_i H.</math>
Given ''y'' in ''G'', each ''yx<sub>i</sub>'' is in some coset ''x<sub>j</sub>H'' and so 
:<math>yx_i = x_jh_i</math>
for some index ''j'' and some element ''h''<sub>''i''</sub> of ''H''. 
The value of the transfer for ''y'' is defined to be the image of the product 
:<math>\textstyle \prod_{i=1}^n h_i </math>
in ''H''/''H''′, where ''H''′ is the commutator subgroup of ''H''. The order of the factors is irrelevant since ''H''/''H''′ is abelian.

It is [[Artin transfer (group theory)#Independence of the transversal|straightforward]] to show that, though the individual ''h<sub>i</sub>'' depends on the choice of coset representatives, the value of the transfer does not. It is also [[Artin transfer (group theory)#Homomorphisms|straightforward]] to show that the mapping defined this way is a homomorphism.

==Example==

If ''G'' is cyclic then the transfer takes any element ''y'' of ''G'' to ''y''<sup>[''G'':''H'']</sup>.

A simple case is that seen in the [[Gauss's lemma (number theory)|Gauss lemma]] on [[Quadratic residue|quadratic residue]]s, which in effect computes the transfer for the multiplicative group of non-zero residue classes modulo a [[Prime number|prime number]] ''p'', with respect to the subgroup {1, &minus;1}.<ref name=S122/>  One advantage of looking at it that way is the ease with which the correct generalisation can be found, for example for cubic residues in the case that ''p'' &minus; 1 is divisible by three.

==Homological interpretation==

This homomorphism may be set in the context of [[Group cohomology#Group homology|group homology]]. In general, given any subgroup ''H'' of ''G'' and any ''G''-module ''A'', there is a corestriction map of homology groups <math>\mathrm{Cor} : H_n(H,A) \to H_n(G,A)</math> induced by the inclusion map <math>i: H \to G</math>, but if we have that ''H'' is of finite index in ''G'', there are also restriction maps <math>\mathrm{Res} : H_n(G,A) \to H_n(H,A)</math>. In the case of ''n ='' 1 and <math>A=\mathbb{Z}</math> with the trivial ''G''-module structure, we have the map <math>\mathrm{Res} : H_1(G,\mathbb{Z}) \to H_1(H,\mathbb{Z})</math>. Noting that <math>H_1(G,\mathbb{Z})</math> may be identified with <math>G/G'</math> where <math>G'</math> is the commutator subgroup, this gives the transfer map via <math>G \xrightarrow{\pi} G/G' \xrightarrow{\mathrm{Res}} H/H'</math>, with <math>\pi</math> denoting the natural projection.<ref name=S120>Serre (1979) p.120</ref>  The transfer is also seen in [[Algebraic topology|algebraic topology]], when it is defined between [[Classifying space|classifying space]]s of groups.

==Terminology==

The name ''transfer'' translates the German ''Verlagerung'', which was coined by [[Biography:Helmut Hasse|Helmut Hasse]].

==Commutator subgroup==

If ''G'' is finitely generated, the [[Commutator subgroup|commutator subgroup]] ''G''′ of ''G'' has finite index in ''G'' and ''H=G''′, then the corresponding transfer map is trivial. In other words, the map sends ''G'' to 0 in the abelianization of ''G''′. This is important in proving the [[Principal ideal theorem|principal ideal theorem]] in [[Class field theory|class field theory]].<ref name=S122>Serre (1979) p.122</ref>  See the [[Biography:Emil Artin|Emil Artin]]-John Tate ''Class Field Theory'' notes.

== See also ==
* [[Focal subgroup theorem]], an important application of transfer
* By Artin's reciprocity law, the [[Artin transfer (group theory)#Artin transfer|Artin transfer]] describes the principalization of ideal classes in extensions of algebraic number fields.

== References ==
{{reflist}}
*{{citation|journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
|year= 1929|volume= 7|issue= 1|pages= 46–51
|title=Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz
|first=Emil|last= Artin|doi=10.1007/BF02941159|s2cid= 121475651}}
*{{citation|journal=Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften
|last=Schur|first= Issai
|title=Neuer Beweis eines Satzes über endliche Gruppen|jfm= 33.0146.01
|pages= 1013–1019 |year=1902}}
*{{cite book | title=Group Theory | first=W.R. | last=Scott | publisher=Dover | year=1987 | isbn=0-486-65377-3 | pages=60 ff | zbl=0641.20001 | orig-year=1964 }}
*{{cite book | last=Serre | first=Jean-Pierre |  title=[[Local Fields]] | translator-link1=Marvin Greenberg |translator-first1=Marvin Jay |translator-last1=Greenberg | series=[[Graduate Texts in Mathematics]] | volume=67 | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | year=1979 | isbn=0-387-90424-7 | zbl=0423.12016 | pages=120–122 }}

[[Category:Group theory]]

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