On intersections of abelian and nilpotent subgroups in finite groups
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 3, pp. 128-131
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Let $A$ be an abelian subgroup of a finite group $G$, and let $B$ be a nilpotent subgroup of $G$. If $G$ is solvable, then we prove that it contains an element $g$ such that $A\bigcap B^g\le F(G)$, where $F(G)$ is the Fitting subgroup of $G$. If $G$ is not solvable, we prove that a counterexample of smallest order to the conjecture that $A\bigcap B^g\le F(G)$ for some element $g$ of $G$ is an almost simple group.
Keywords:
finite group, abelian subgroup, nilpotent subgroup, intersection of subgroups, fitting subgroup.
@article{TIMM_2015_21_3_a13,
author = {V. I. Zenkov},
title = {On intersections of abelian and nilpotent subgroups in finite groups},
journal = {Trudy Instituta matematiki i mehaniki},
pages = {128--131},
publisher = {mathdoc},
volume = {21},
number = {3},
year = {2015},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TIMM_2015_21_3_a13/}
}
V. I. Zenkov. On intersections of abelian and nilpotent subgroups in finite groups. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 3, pp. 128-131. http://geodesic.mathdoc.fr/item/TIMM_2015_21_3_a13/