Source algebras and cohomology algebras of block ideals of finite groups with defect groups isomorphic to extraspecial \(p\)-groups (Q6634542)

Let \(p\) be a prime number and \(k\) an algebraically closed field of characteristic \(p\). Let \(G\) be a finite group with \(p \in \pi(G)\) and consider the group algebra \(kG\) as a left \(k[G\times G]\)-module by posing \((x,y)\alpha=x\alpha y^{-1}\) for all \(x,y \in G\) and \(\alpha \in kG\). Let \((P,e_{P})\) be a maximal \(b\)-Brauer pair such that \(\mathrm{Br}_{P}(i)e_{P} = \mathrm{Br}_{P}(i)\) and let \(X\) be the source algebra of the block ideal \(b\) with \(i\), namely \(X=ikGi\). One can view \(X\) as a \((kP,kP)\)-bimodule. The author is interested in the \((kP,kP)\)-bimodule structure of \(X\).\N\NIn this paper, the author intends to examine block ideals with defect groups isomorphic to extraspecial \(p\)-groups of order \(p^{3}\) and exponent \(p\). The author first analyzes bimodule structures of these block ideals and provides a direct sum decomposition. He then proves that the images of the transfer maps on the cohomology rings of defect groups, defined by the source algebras, coincide with the cohomology rings of the block ideals in question.