Editing Transfer (group theory) (section)

==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]], when it is defined between [[classifying space]]s of groups.