Index of the center of an irreducible nilpotent linear group over an algebraically closed field (Q788841)

A well-known theorem of D. A. Suprunenko says that the index of the center in an irreducible s-step nilpotent group consisting of matrices of degree n over an algebraically closed field is finite and does not exceed the number \(n!n^{(n-1)(s-1)}\) [see, for example, \textit{D. A. Suprunenko}, Matrix groups (1972; Zbl 0253.20074), p. 294]. For square- free degrees n, D. A. Suprunenko proved that this index does not exceed the number \(n^ s\), where this estimate is sharp. Let p be a prime, k be an integer, and \(\Omega\) be an algebraically closed field. It is proved in this article that there exists an irreducible s-step nilpotent group of matrices of degree \(n=p^ k\) over \(\Omega\) such that the index of its center is greater than \(n^ s\), if \(k>1\), and is equal to \(n^ s\) for \(k=1\) (Theorem 1). Similar groups exist also for certain non-primary degrees n, which are not square-free (Theorem 2). A sharp estimate of the index of the center is given for \(n=4\). Namely, let G be an irreducible s-step nilpotent group, \(s\geq 8\), consisting of matrices of degree 4 over \(\Omega\), and let Z be its center. Then \(| G:Z| \leq 2^{3s-2}.\) On the other hand, for any integer \(s\geq 3\) there exists an irreducible s-step nilpotent group \(G\leq GL_ 4(\Omega)\) such that \(| G:Z| = 2^{3s-2}\) (Theorem 3).