A derived \(\pi\)-length of \(\pi\)-soluble groups and central \(\pi\)-Hall intersections
O. Shpyrko
Only finite groups are considered. Let \(G\) be a \(\pi\)-soluble group.
The smallest natural number of abelian \(\pi\)-factors among all subnormal
series of group \(G\) with \(\pi'\)-factors or abelian \(\pi\)-factors is called
a derived \(\pi\)-length of \(G\) and is defined by \(l_{\pi}^a(G).\)
It is clear that
\(l_{\pi}(G)\le l_{\pi}^n(G) \le l_{\pi}^a(G)\) for all \(\pi\)-soluble group
\(G\) and, in the case \(\pi=\{p\}\), we have
\(l_{\pi}^a(G)=l_{\pi}^n(G)=l_{\pi}(G)=l_p(G).\)
Recall that the central intersection of \(\pi\)-Hall subgroups means the
intersection of two different \(\pi\)-Hall subgroups containing the center
of one of them.
In [1-3], estimates were provided for the \(p\)-length \(l_p(G)\) of
a \(p\)-soluble group \(G\) (the \(\pi\)-length \(l_{\pi}(G)\) and the nilpotent \(\pi\)-length
\(l_{\pi}^{n}(G)\) of a \(\pi\)-soluble group \(G\)) depending on the structure
of central \(p\)-Sylow (\(\pi\)-Hall) intersections.
Similar results were obtained for the derived \(\pi\)-length of
\(\pi\)-soluble groups.
Theorem.\(\ 1.\ \) If the \(\pi\)-soluble group \(G\)
has no central \(\pi\)-Hall intersection then
\(l_{\pi}^a(G)=d(G_{\pi}).\)
\(\ \ 2.\ \) If, in a \(\pi\)-soluble group \(G\), the central \(\pi\)-Hall intersections
are abelian either Schmidt group then \(l_{\pi}^a(G)\le 1+d(G_{\pi}).\)
Corollary 1.If the \(p\)-soluble group \(G\)
has no central \(p\)-Sylow intersection then
\(l_{p}^a(G)=d(G_{p}).\)
Corollary 2.Let \(G\) be a \(\pi\)-soluble group with metabelian
central \(\pi\)-Hall intersections. Then \(l_{\pi}^a(G)\le 3.\)
Corollary 3. Let \(G\) be a \(\pi\)-soluble group with biabelian
central \(\pi\)-Hall intersections. Then \(l_{\pi}^a(G)\le 3.\)
References
A. G. Anizhenko, V. S. Monakhov, Central intersections and
\(p\)-length of \(p\)-soluble groups, Dokl. AS BSSR, 21, N11 (1977), 968–971.
S. V. Putilov, About normal structure of finite \(\pi\)-soluble groups,
Voprosy algebry, 5, University, Minsk (1990), 82–86.
O. A. Shpyrko, About intersections Hall subgroups and \(\pi\)-length of
\(\pi\)-soluble group, VSU Vestnik, 4, N18 (2000), 82–89.
