Transfer and Tate's theorem.
For a prime \(p\), let \(E^p(X)\), \(A^p(X)\), and \(O^p(X)\) denote, respectively, the smallest normal subgroup of a finite group \(X\) such that the quotient is an elementary Abelian \(p\)-group, an Abelian \(p\)-group, and a \(p\)-group. \textit{J. Tate}'s theorem [Topology 3, Suppl. 1, 109-111 (1964; Zbl 0125.01503)] states that for a finite group \(G\) and a subgroup \(H\leq G\) containing a Sylow \(p\)-subgroup of \(G\) the following are equivalent: (1)~\(E^p(G)\cap H=E^p(H)\); (2)~\(A^p(G)\cap H=A^p(H)\); (3)~\(O^p(G)\cap H=O^p(H)\). Tate's proof was cohomological, and Thompson gave a character-theoretic proof (which can be found in [\textit{I. M. Isaacs}, Character theory of finite groups. Corrected reprint of the 1976 original. Providence: AMS Chelsea Publishing (2006; Zbl 1119.20005)]). In this note the authors produce a proof which is elementary in the sense that it is based on transfer theory. By the same method, it is also proved that if a subgroup \(Y\) of a finite group \(X\) contains a Sylow \(p\)-subgroup of \(X\), then \(X/A^p(X)\) is isomorphic to a direct factor of \(Y/A^p(Y)\).
scientific article; zbMATH DE number 5500770