학술논문
On a generalization of a theorem of Burnside
Document Type
Text
Author
Source
Czechoslovak Mathematical Journal | 2015 Volume:65 | Number:3
Subject
Language
English
Abstract
Jiangtao Shi.
Obsahuje seznam literatury
A theorem of Burnside asserts that a finite group G is p-nilpotent if for some prime p a Sylow p-subgroup of G lies in the center of its normalizer. In this paper, let G be a finite group and p the smallest prime divisor of |G|, the order of G. Let P \in Syl_{p} (G). As a generalization of Burnside’s theorem, it is shown that if every non-cyclic p-subgroup of G is self-normalizing or normal in G then G is solvable. In particular, if P \not\cong \left\langle {a,b;{a^{{p^{n - 1}}}} = 1,{b^2} = 1,{b^{ - 1}}ab = {a^{1 + {p^{n - 2}}}}} \right\rangle, where n\geq 3 for p > 2 and n\geq 4 for p = 2, then G is p-nilpotent or p-closed.
Obsahuje seznam literatury
A theorem of Burnside asserts that a finite group G is p-nilpotent if for some prime p a Sylow p-subgroup of G lies in the center of its normalizer. In this paper, let G be a finite group and p the smallest prime divisor of |G|, the order of G. Let P \in Syl_{p} (G). As a generalization of Burnside’s theorem, it is shown that if every non-cyclic p-subgroup of G is self-normalizing or normal in G then G is solvable. In particular, if P \not\cong \left\langle {a,b;{a^{{p^{n - 1}}}} = 1,{b^2} = 1,{b^{ - 1}}ab = {a^{1 + {p^{n - 2}}}}} \right\rangle, where n\geq 3 for p > 2 and n\geq 4 for p = 2, then G is p-nilpotent or p-closed.