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 \(l(G') \leq 2\).
scientific article; zbMATH DE number 554381