Finite pgroups with a Frobenius group of automorphisms whose kernel is a cyclic pgroup
Abstract
Suppose that a finite $p$group $P$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ that is a cyclic $p$group and with complement $H$. It is proved that if the fixedpoint subgroup $C_P(H)$ of the complement is nilpotent of class $c$, then $P$ has a characteristic subgroup of index bounded in terms of $c$, $C_P(F)$, and $F$ whose nilpotency class is bounded in terms of $c$ and $H$ only. Examples show that the condition of $F$ being cyclic is essential. The proof is based on a Lie ring method and a theorem of the authors and P. Shumyatsky about Lie rings with a metacyclic Frobenius group of automorphisms $FH$. It is also proved that $P$ has a characteristic subgroup of $(C_P(F), F)$bounded index whose order and rank are bounded in terms of $H$ and the order and rank of $C_P(H)$, respectively, and whose exponent is bounded in terms of the exponent of $C_P(H)$.
 Publication:

arXiv eprints
 Pub Date:
 February 2013
 DOI:
 10.48550/arXiv.1302.3499
 arXiv:
 arXiv:1302.3499
 Bibcode:
 2013arXiv1302.3499K
 Keywords:

 Mathematics  Group Theory;
 17B40;
 17B70;
 20D15
 EPrint:
 references updated, a few typos corrected. arXiv admin note: text overlap with arXiv:1301.3409