Pretransfer retains the actual product inside the target subgroup before passing to its derived quotient. It depends on a transversal and need not be a homomorphism, but it is the right object for orbit and double-coset calculations. [@isaacs2008]
Definition
If \(H\) is nonabelian, an ordering of the factors is required. Changing the transversal or order can change \(V_T(g)\), but the change vanishes modulo \(H'\).
Core result
When comparing \(V_T(xy)\), the intermediate transversal permutation reorders and pairs factors modulo \(H'\), proving multiplicativity. A change of transversal introduces conjugate correction factors whose product vanishes in the abelianization. See Section 5A.
Structural properties
- Pretransfer itself is generally not a homomorphism and its target need not be abelian.
- Grouping the transversal by cyclic-subgroup orbits yields power and conjugate-product evaluation formulas.
- Across a normal subgroup of index \(p\), pretransfer reduces modulo the derived subgroup to a norm or a \(p\)th power.
- For nested subgroups, composed pretransfers agree modulo the smallest target's derived subgroup with direct pretransfer.
- Yoshida's theorem detects a wreath-product quotient when pretransfer escapes a Frattini subgroup.
Transfer is canonical, while pretransfer is a noncanonical lift used in proofs. The lift preserves local products that disappear after abelianization.
Example and boundary
Knowledge network
The transfer core article supplies the full definition, while transitivity and Yoshida's theorem show why pretransfer is retained in advanced calculations.
Proof and sources
Definition, transversal independence, and orbit evaluation are in Section 5A; index-\(p\) and double-coset calculations are in Section 10A. [@isaacs2008]
