 Transfer (group theory)

For a subgroup $H$ of finite index, choose representatives $t$ of right cosets $Ht$. Write $tg=h(t,g)t_g$ with $h(t,g)\in H$. The transfer is $V_{G,H}(g)={\prod }_{t}h(t,g)(\mathrm{mod}{H}^{\prime })$. Replacing $t$ by $a_{t}t$ multiplies each factor by $a_t$ on the left and $a_{t_g}^{-1}$ on the right; these cancel in the abelianization because $t\mapsto t_g$ is a permutation. The identity $h(t,g_1{g}_2)=h(t,g_1)h(t_{g_1},g_2)$ proves the group homomorphism property after multiplying all factors.

In group theory, "transfer" typically refers to a specific concept related to the behavior of groups under certain conditions, particularly in the context of transfer homomorphisms or transfer maps which can arise in the study of group cohomology and modular representations. However, in a more specific context, "transfer" often relates to the idea of transferring properties or structures from one group to another.