Elementary abelian operator groups
Copyright policy )Elementary abelian operator groups
Let \(A\) be an elementary abelian \(p\)-group which acts on a solvable \(p'\)- group \(G\). If \(\phi \in A\), \(C_ G(\phi)\) denotes the fixed point subgroup of \(\phi\). Previous results of \textit{A. Turull} [J. Algebra 86, 555-566 (1984; Zbl 0526.20017)] and of \textit{F. Gross} [Bull. Aust. Math. Soc. 7, 91-100 (1972; Zbl 0241.20016)] indicate that there is a strong connection between the nilpotency length \(l(G)\) of \(G\) and the nilpotency length of the fixed point subgroups of the nontrivial elements of \(A\). The paper contains two main results: 1) Let \(n\) be a positive integer and let \(G\) be a finite solvable \(p'\)-group acted upon by an elementary abelian \(p\)-group \(A\) of order \(p^{n + 1}\). If \(l(C_ G(\phi)) \leq n\) for every nontrivial element \(\phi\) of \(A\), then \(l(G) \leq n + 1\). This theorem shows that the results of Turull and Gross (loc. cit.) provide the best bounds for \(l(G)\) in their cases. 2) Let \(G\) be a periodic solvable \(2'\)-group acted upon by an elementary abelian group \(A\) of order 8. Suppose that \(C_ G(\phi)\) has a nilpotent derived subgroup for every nontrivial element \(\phi\) of \(A\). Then \(l(G') \leq 2\).
- DigiZeitschriften Germany
- Technion – Israel Institute of Technology Israel
nilpotent derived subgroup, periodic solvable \(2'\)-group, Solvable groups, supersolvable groups, Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks, finite solvable groups, Article, Automorphisms of infinite groups, 510.mathematics, elementary abelian \(p\)-group, Derived series, central series, and generalizations for groups, nilpotency length, Periodic groups; locally finite groups, fixed point subgroup, Fitting length, Arithmetic and combinatorial problems involving abstract finite groups
nilpotent derived subgroup, periodic solvable \(2'\)-group, Solvable groups, supersolvable groups, Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks, finite solvable groups, Article, Automorphisms of infinite groups, 510.mathematics, elementary abelian \(p\)-group, Derived series, central series, and generalizations for groups, nilpotency length, Periodic groups; locally finite groups, fixed point subgroup, Fitting length, Arithmetic and combinatorial problems involving abstract finite groups
citations This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).2 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Average influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Average impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Average
Citations provided by BIP!Popularity provided by BIP!