Some special coprime actions and their consequences. (English) Zbl 07905696
Let a group \(A\) act on the group \(G\) coprimely. The authors determine bounds for the \(p\)-length of \(G\) in terms of the order of \(A.\) In particular, denoting with \(l(A:B)\) the length of the longest chain of subgroups of \(A\) that starts with \(B\) and ends with \(A\), the authors prove: let \(A\) be a group acting coprimely and with regular orbits on the solvable group \(G\) and suppose that \(B\) is a subgroup of \(A\) such that \(\cap_{a\in A}[G,B]^a=1;\) if \(p\) is a prime not dividing \(|C_G(A)|,\) then \(l_p(G)\leq l(A:B).\) Moreover it is investigated how some \(A\)-invariant \(p\)-subgroups are embedded in \(G\). In particular, suppose that \(A\) is abelian and that \(G\) contains a unique \(A\)-invariant Sylow \(p\)-subgroup \(P\) where \(C_G(A)\) is a \(p^\prime\)-group; if \(G\) is \(p\)-solvable then \(l_p(G)\leq l(A)\) and if \(G\) is solvable and \(p\) is odd then \(P\leq F_{2l(A)}(G).\)
Reviewer: Andrea Lucchini (Padova)
MSC:
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 20D45 | Automorphisms of abstract finite groups |
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