All considered groups are finite. In [1], R. Baer studied properties of the hypercenter \(Z_\infty(G)\):
Theorem 1. [1] For any group \(G\), the following conditions hold.
\(\ \ (1)\) If \(p\) is a prime then a \(p\)-element \(g\in Z_\infty(G)\) iff \(g\) permutes with all elements of \(G\) whose orders are coprime to the order of \(g\).
\(\ \ (2)\) The hypercenter of \(G\) is the intersection of normalizers of all Sylow subgroups of
\(G\).
The following class is a natural generalization of nilpotent groups. Let \(\pi\) be a nonempty set of primes. Assume that \(\{ \pi_i | i\in I, i\neq j \Rightarrow \pi_i\cap\pi_j=\emptyset\}\) is a partition of \(\pi\). Denote by \(\underset{i\in I}{\times}\mathfrak{G_{\pi_i}}\) the class of all groups that are the direct products of their Hall \(\pi_i\)-subgroups. This class is a particular case of saturated lattice formations, see [2] or [3]. The main goal of this work is to extend Baer's result for such classes.
According to [4, p.389], for a local formation \(\mathfrak{F}\), the \(\mathfrak{F}\)-hypercenter \(Z_\mathfrak{F}(G)\) of a group \(G\) is the maximal \(F\)-hypercentral normal subgroup for the canonical local definition \(F\) of \(\mathfrak{F}\). It was shown that, for any group \(G\), the \(\mathfrak{F}\)-hypercenter exists and is unique. In particular, when \(\mathfrak{F}\) is the class of all nilpotent groups, we have \(Z_\infty(G)=Z_\mathfrak{F}(G)\).
Our main result is
Theorem 2.
Let \(\sigma=\{ \pi_i | i\in I\) and \( \pi_i\cap\pi_j=\emptyset\) for all \(i\neq j\}\) be a partition of a nonempty set of primes \(\pi\) and let \(\mathfrak{F}=\underset{i\in I}{\times}\mathfrak{G_{\pi_i}}\). If \(G\) is a \(\pi\)-group then:
\(\ \ (1)\) A \(\pi_i\)-element \(g\) of \(G\) belongs to \(Z_\mathfrak{F}(G)\) iff \(g\) permutes with all \(\pi_i'\)-elements of \(G\).
\(\ \ (2)\) The intersection of normalizers of all maximal \(\pi_i\)-subgroups of \(G\) for all \(i\in I\) is \(Z_\mathfrak{F}(G)\).
References
R. Baer, Group elements of prime power Index, Trans. Amer. Math Soc., 75, N1 (1953), 20–47.
A. F. Vasil'ev, S. F. Kamornikov, On the Kegel-Shemetkov problem on lattices of generalized subnormal subgroups of finite groups, Algebra Logika, 41 N4, (2002), 411–428.
A. Ballester-Bolinches, K. Doerk, M.D. Perez-Ramos, On the lattice of \(\mathfrak{F}\)-subnormal subgroups, J. Algebra, 148, N1 (1992), 42–52.
K. Doerk, T. Hawkes, Finite soluble groups, Walter de Gruyter, Berlin–New York (1992), 891 p.
