Block induction for group algebras over semi-local rings
The paper continues the development of the theory of \(\pi\)-blocks, where \(\pi\) is a set of primes. The work in this direction began with the papers of \textit{G. R. Robinson} [J. Algebra 117, 409-418 (1988; Zbl 0648.20008)] and \textit{G. R. Robinson} and \textit{R. Staszewski} [ibid. 126, 310-326 (1989; Zbl 0684.20009)]. The analogue of Brauer's Second Main Theorem has been proved in Robinson's paper and Robinson and Staszewski studied subpairs. For \(\pi\)-separable groups the theory of \(\pi\)-blocks is very similar to the theory of \(p\)-blocks. Such a theory was developed by \textit{M. Slattery} [ibid. 102, 60-77 (1986; Zbl 0595.20014); ibid. 124, 236-269 (1989; Zbl 0684.20008)]. The author deals with block induction. Let \(G\) be a finite group and \(R\) be a principal ideal domain containing a primitive \(|G|\)th root of unity and having quotient field of characteristic zero. Let \(p_1,\dots,p_r\) be primes, \(\pi=\{p_1,\dots,p_r\}\). Suppose that \(R\) contains a unique prime ideal \({\mathcal P}_i\) such that \(p_i\in{\mathcal P}_i\), \(i =1,\dots,r\). The group \(G\times G\) acts on \(RG\) by \(\alpha (x,y)=x^{-1}\alpha y\), \(\alpha\in RG\), \(x,y\in G\). The \(\pi\)- blocks are the indecomposable summands of the \(R(G\times G)\)-module \(RG\). The main results are: Theorem 1. Let \(D\) be a nilpotent \(\pi\)-subgroup of \(G\), and let \(b\) be a \(\pi\)-block of \(L=DC_G(D)\). Then \(b^G\) is the unique block of \(G\) satisfying the following conditions: (1) \(b|(b^G)_{L\times L}\): (2) If \(p\in \pi\) and \(P\) is a Sylow \(p\)-subgroup of \(D\), then \((b^{PC_G(D)})^G=b^G\). Theorem 2. Suppose that \(G\) is a group with nilpotent Hall \(\pi\)- subgroup. Let \(B\) be a block of \(G\), and let \({\mathcal L}_G(B)=\{(D_i,b^*_i):i\in I\}\) be a set of representatives for the conjugacy classes of maximal \(B\)-subpairs of \(G\). Then \(B|\bigoplus_{i\in I}b_i^{G\times G}\).
scientific article; zbMATH DE number 614241