The Steinberg component of a finite group with a split (B,N)-pair
Let \(G=(G,B,N,R,U)\) be a finite group with an unsaturated split (B,N)- pair of characteristic p (prime) and rank n [see \textit{C. W. Curtis}, Lect. Notes Math. 131, B1-B39 (1970; Zbl 0233.20005) p. B-8]. Here \(B=UH\) is a semidirect product where U is a normal p-subgroup of B and H is abelian with order prime to p. The Weyl group \(W=N/H\) is generated by the set \(R=\{w_ 1,...,w_ n\}\) of involutions. Let \(\cap_{n\in N}n^{- 1}Un\neq \{1\}\). Let \(\nu\) : \(N\to W\) be the natural epimorphism with \(\nu^{-1}(w)=(w)\) for \(w\in W\). Let F be an algebraically closed field of characteristic r (prime or zero). Let \(Y=Ind^ G_ U(F_ U)\) where \(F_ U\) is F treated as a trivial U-module and let \([T]=\sum_{t\in T}\) \(t\in FG\) for any subset T of G. In [Can. J. Math. 9, 347-351 (1957; Zbl 0079.256)] \textit{R. Steinberg} defined the module FGe where \(e=\sum_{w\in W}(-1)^{\ell (w)}(w)[B]\) and proved that FGe is simple when \(r=0\) or r does not divide \(| G:B|\). In this paper the author finds the socle of the Steinberg module in arbitrary characteristic by studying the indecomposable components of Y; in particular, by examining what he calls the Steinberg component.
finite group with an unsaturated split (B,N)-pair of characteristic p
scientific article; zbMATH DE number 4029829