The Transfer Maps for the Cohomology Rings of Block Ideals with Wreathed Defect Groups

Abstract

Let G be a finite group. We investigate the cohomology ring \(H^*(G,b;A)\) of a block ideal b of the finite group algebra kG, where k is an algebraically close field of characteristic p and A is a source algebra of b. The author expected in Sasaki (Algebr. Represent. Theory 16, 1039–1049, 2013) that, P being its defect group, the image of the transfer map \(t_A\) of the cohomology ring \(H^*(P,k)\) induced by A coincides with \(H^*(G,b;A)\): \(\text {Im} t_A=H^*(G,b;A)\). This expectation has previously been settled in Sasaki (Hokkaido Math. J. 53, 443–462, 2024) for blocks of tame representation type and blocks with extraspecial p-groups as P. In this note, we prove that this expectation holds for b whose defect group P is isomorphic to a wreathed 2-group. The proof relies on prior works Okuyama and Sasaki (Algebr. Represent. Theory 4, 405–444, 2001; J. Algebra 497, 92–101, 2018), Kawai and Sasaki (J. Algebra 306(2), 301–321, 2006), and Sasaki (J. Algebra 666, 777–793, 2025).