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In the mathematical field of [[group theory]], the '''transfer''' defines, given a [[Group (mathematics)|group]] ''G'' and a [[subgroup]] of finite [[Index of a subgroup|index]] ''H'', a [[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 [[Issai Schur]] and rediscovered by [[Emil Artin]].<ref name=S122/>


==Construction==
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The construction of the map proceeds as follows:<ref>Following Scott 3.5</ref> Let [''G'':''H''] = ''n'' and select [[coset]] representatives, say
:<math>x_1, \dots, x_n,\,</math>
for ''H'' in ''G'', so ''G'' can be written (disjoint union)
:<math>G = \dot{\cup}\ 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</math>
for some index ''j'' and some element ''h'' of ''H'', Then in general
:<math>yx_i = x_{iA}h_i</math>
where ''A''=''A''(''y'') is some mapping of {1,2,…,''n''} to itself and each ''h<sub>i</sub>''=''h<sub>i</sub>''(''y'') is an element of ''H''.
 
The value of the transfer for ''y'' is defined to be the product in ''H''/''H''′
:<math>\textstyle \prod_{i=1}^n h_i H'</math>
where ''H''′ is the commutator subgroup of ''H''. Note that the order of the factors is irrelevant since ''H''/''H''′ is abelian.
 
It's 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's also straightforward to show that the mapping defined this way is a homomorphism.
 
==Example==
 
A simple case is that seen in the [[Gauss's lemma (number theory)|Gauss lemma]] on [[quadratic residue]]s, which in effect computes the transfer for the multiplicative group of non-zero [[residue class]]es modulo a [[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]] (strictly, group ''homology''), providing a more abstract definition.<ref name=S120>Serre (1979) p.120</ref> The transfer is also seen in [[algebraic topology]], when it is defined between [[classifying space]]s of groups.
 
==Terminology==
 
The name ''transfer'' translates the German ''Verlagerung'', which was coined by [[Helmut Hasse]].
 
==Commutator subgroup==
 
If ''H'' is the [[commutator subgroup]] ''G''′ of ''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]] in [[class field theory]].<ref name=S122>Serre (1979) p.122</ref>  See the [[Emil Artin]]-[[John Tate]] ''Class Field Theory'' notes.
 
== See also ==
* [[Focal subgroup theorem]], an important application of transfer
 
== References ==
{{reflist}}
*{{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 | origyear=1964 }}
*{{cite book | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=Local fields | others=Translated from the French by Marvin Jay Greenberg | series=[[Graduate Texts in Mathematics]] | volume=67 | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-387-90424-7 | zbl=0423.12016 | pages=120–122 }}
 
[[Category:Group theory]]

Latest revision as of 09:28, 8 January 2015


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