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On equations in finite groups and invariants of representations for their subgroups. (English) Zbl 0854.20011

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 \(g\in G\).

MSC:

20C20 Modular representations and characters
20C15 Ordinary representations and characters
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20F12 Commutator calculus
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