Mackey transfer theorem

Mackey's formula describes transfer from \(G\) to \(H\) when restricted to another subgroup \(K\), decomposing it over \((H,K)\) double cosets. ^{[@isaacs2008]}

Definition

Conjugation carries each local value back into \(H\), and the final product is taken in \(H/H'\).

Core result

Partition the right cosets of \(H\) into \(K\)-orbits. The orbits correspond to double cosets and their stabilizers are conjugates of \(K\cap H^x\). Orbit evaluation yields the product of local pretransfers.

Structural properties

• It is the group-transfer counterpart of Mackey decomposition in representation theory.
• Each double coset contributes one local intersection term.
• When \(K\leq H\), it reduces to a simpler restriction formula.
• Together with transitivity, local terms can be decomposed further along subgroup chains.
• Yoshida uses it to compare global transfer with index-\(p\) transfer inside a Sylow subgroup.

The formula organizes two nonnested subgroups through double cosets instead of manipulating one enormous transversal directly.

Example and boundary

Knowledge network

Pretransfer supplies local terms, transitivity handles nested directions, and Yoshida is the central application.

• Pretransfer
• Transitivity of transfer
• Yoshida transfer theorem

Proof and sources

The double-coset orbit proof and formula are Theorem 10.10. ^{[@isaacs2008]}