Theorem. Finite-index transfer [frap-000W]
✍️source
MonoidHom.transferMonoidHom.transfer_def
L∃∀N
transfer_eq_prod_of_bijective
L∃∀N

    Let \(H\leq G\) have finite index. Choose representatives \(r_i\) for the left cosets \(r_iH\), indexed bijectively by a finite set \(I\). For \(g\in G\), left multiplication permutes the cosets, so uniquely write

    \[g r_i=r_{\pi _g(i)}h_i(g),\qquad h_i(g)\in H.\]

    Write \(H_{\mathrm {ab}}=H/[H,H]\) and \([h]\) for the class of \(h\in H\). The abelianization is the target of the transfer

    \[T_{G,H}:G\longrightarrow H_{\mathrm {ab}},\qquad T_{G,H}(g)=\prod _{i\in I}[h_i(g)]\]

    is a homomorphism independent of the chosen representatives. This is the left-coset construction of [mayer2015artin, Definition 3.1, Proposition 3.1 and Theorem 3.1, pp. 4--6]. Neither finiteness of \(G\) nor normality of \(H\) is required.

    Proof. 

      We compose left actions rightmost first. Multiplication on cosets gives \(\pi _{gk}=\pi _g\pi _k\). Expanding \(gkr_i\) in two steps yields

      \[h_i(gk)=h_{\pi _k(i)}(g)h_i(k).\]

      Taking classes and multiplying over \(i\) gives \(T_{G,H}(gk)=T_{G,H}(g)T_{G,H}(k)\), because \(\pi _k\) reorders the first factors. Replace \(r_i\) by \(r_i t_i\), with \(t_i\in H\). The new residual factor is \(t_{\pi _g(i)}^{-1}h_i(g)t_i\). Its total class contains each \([t_i]\) and its inverse exactly once, so they cancel. Reindexing the cosets also only reorders an abelian product. This proves independence of all choices.

      The linked transfer definitions specialize the homomorphism \(\phi :H\to A\) to the quotient \(H\to H_{\mathrm {ab}}\). The indexed formula takes \(f(i)=r_i\) and \(\pi =\pi _g\); its factor \(r_{\pi _g(i)}^{-1}g r_i\) is precisely \(h_i(g)\). Its finite-index instance follows from the finite bijective family of cosets.