= Transfer
{disambiguate=group theory}
{title2=$V_{G,H}:G\longrightarrow H/H'$}
{wiki}
= Transfer homomorphism
{synonym}
= Group transfer
{synonym}
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 $V_{G,H}(g)=\prod_t h(t,g)\pmod{H'}$. Replacing $t$ by $a_tt$ 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 $h(t,g_1g_2)=h(t,g_1)h(t_{g_1},g_2)$ proves the <group homomorphism> property after multiplying all factors.
Back to article page
