Transfer homomorphism for a finite index subgroup

Let $G$ be a finite group, let $H\le G$ be a subgroup, let $A$ be an abelian group written multiplicatively, and let $\phi :H\to A$ be a group homomorphism (Monoid homomorphism and group homomorphism). Write $H\mathrm{\backslash }G$ for the set of right cosets $Ht$ of $H$ in $G$ (Left and right cosets $gH$ and $Hg$ of a subgroup), and for each coset $\alpha =H{t}_{\alpha }\in H\mathrm{\backslash }G$ choose a representative $t_{\alpha }$.

The transfer. For $x\in G$ the transfer of $\phi$ is

$V_{\phi }(x):={\prod }_{\alpha \in H\mathrm{\backslash }G}\phi \text{\hspace{0.17em}⁣}\left(t_{\alpha }x{t}_{\alpha x}^{-1}\right),\text{where }\alpha x:=H{t}_{\alpha }x\mathrm{.}$

Two comments make the displayed formula a definition rather than a shorthand.

1. Each factor lies in $H$, so $\phi$ is evaluated inside its domain. The coset $\alpha x=H{t}_{\alpha }x$ is a right coset of $H$, with the chosen representative $t_{\alpha x}$; hence $t_{\alpha }x{t}_{\alpha x}^{-1}\in H$, because $t_{\alpha }x$ and $t_{\alpha x}$ represent the same right coset ($x\in aH$ iff $a^{-1}x\in H$, and $aH=bH$ iff $a^{-1}b\in H$).
2. The product is well defined although the coset set is unordered. The index set $H\mathrm{\backslash }G$ is finite, and $A$ is abelian, so the product of the finitely many elements $\phi (t_{\alpha }x{t}_{\alpha x}^{-1})\in A$ does not depend on the order in which the factors are written; the factor attached to the coset $\alpha$ is determined by $\alpha$, $x$, the chosen representatives and $\phi$.

The definition of $V_{\phi }$ therefore depends on the chosen transversal $\{t_{\alpha }\}$; the fact that it does not depend on that choice is proved in Transfer is independent of the transversal, and the fact that $V_{\phi }:G\to A$ is a homomorphism is proved in Transfer is a homomorphism.

• Right action convention. The symbol $\alpha x$ denotes right multiplication on the coset, $\alpha x=H{t}_{\alpha }\cdot x$, and the coset assignment $\alpha \mapsto \alpha x$ is a permutation of the finite set $H\mathrm{\backslash }G$. The transfer is thus built from the right action of $G$ on its right cosets of $H$, the choice that makes every factor $t_{\alpha }x{t}_{\alpha x}^{-1}$ lie in $H$; with left cosets the analogous factor is $t_{x\alpha }^{-1}x{t}_{\alpha }$, where $\alpha =t_{\alpha }H$ and $x\alpha =x{t}_{\alpha }H=t_{x\alpha }H$.

• Abelian target. Abelianness of $A$ is used only to make the ordering of the product irrelevant; the elements $\phi (\cdot )$ need not commute in a nonabelian target, and the construction is not made there.

• Group-theoretic role. The transfer is the tool that converts information about the $p$-part of $G$ into a homomorphism into an abelian quotient of a Sylow subgroup; it is applied in Burnside normal p complement theorem and Fusion control forces trivial Sylow intersection with the p residual.