Transitivity of transfer

Transitivity says that transfer from \(G\) to \(H\) may be computed through an intermediate subgroup \(K\), analogous to transitivity of a norm. ^{[@isaacs2008]}

Definition

Pretransfers depend on transversals, so the canonical equality appears modulo \(H'\).

Core result

Choose a transversal for \(K\) in \(G\) and one for \(H\) in \(K\); their products form a transversal for \(H\) in \(G\). Expanding kernel components and reordering in \(H/H'\) gives the formula.

Structural properties

• A high-index transfer can be decomposed into index-\(p\) steps.
• The result parallels composition of restriction/corestriction in cohomology.
• At pretransfer level it is valid only modulo the derived subgroup.
• Yoshida's proof applies it along index-\(p\) subgroups of a Sylow group.
• It iterates along an arbitrary finite subgroup chain.

Transitivity makes transfer compatible with recursive calculation along normal or subgroup series.

Example and boundary

Knowledge network

Pretransfer supplies the maps, Mackey handles transverse subgroup directions, and Yoshida uses index-\(p\) decomposition.

• Pretransfer
• Mackey transfer formula
• Yoshida transfer theorem

Proof and sources

The product-transversal proof and exact congruence are Theorem 10.8. ^{[@isaacs2008]}