On equations in finite groups and invariants of representations for their subgroups
This paper is devoted to certain questions in group representation theory from the viewpoint of the theory of equations of groups. The main part is dedicated to the study of the existence of \(p\)-blocks of defect 0. Let \(G\) be a finite group, let a prime \(p\) divide \(|G|\) and let \(\chi\) be an irreducible complex character of \(G\). If \(|G|=p^cm\), \(\chi(1)=p^{c_1}m_1\) with \((m,p)=(m_1,p)=1\), then the nonnegative integer \(c-c_1\) is called the \(p\)-defect of \(\chi\). To give the flavour of the subject, we quote the following result and its corollary. Theorem. Let \(f(x_1,\dots,x_k,u_1,\dots,u_e)\) be a function on a finite group \(G\) and let \(f\) be a product of elementary functions \([x_i,x_{i+1}]\) and \(u^{x_s}_j\) (\(x_i\in G\), \(u_j\in G_p\), \(G_p\) is a Sylow \(p\)-subgroup of \(G\)). Moreover, variables that enter in different elementary factors, are distinct and \(k \geq 2\). Then \(G\) has a \(p\)-block of defect 0 if and only if for some \(g\in G\) the number of solutions of the equation \(f(x_1,\dots,x_k,u_1,\dots,u_e)=g\) does not divide \(p |G_p|^e\). Corollary. The following conditions are equivalent: (i) A finite group \(G\) has a \(p\)-block of defect 0. (ii) The number of solutions of the equation \([x,y]=g\) is prime to \(p\) for some \(g\in G\) \((x,y\in G)\). (iii) The number of solutions of the equation \(u^x v^y=g\) does not divide \(p|G_p|^2\) for some \(g\in G\) (\(x,y\in G\), \(u,v\in G_p\)). (iv) Let \(k\) be a natural number. Then the number of solutions of the equation \([x_1,x_2][x_3,x_4]\cdots[x_{2k-1},x_{2k}]=g\) (\(x_1,\dots,x_{2k}\in G\)) is prime to \(p\) for some \(g\in G\).
scientific article; zbMATH DE number 908495
scientific article