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- Title
Maximal subclasses of local fitting classes.
- Authors
Savelyeva, N. V.; Vorob'ev, N. T.
- Abstract
A Fitting class $$ \mathfrak{F} $$ is said to be π-maximal if $$ \mathfrak{F} $$ is an inclusion maximal subclass of the Fitting class $$ \mathfrak{S}_\pi $$ of all finite soluble π-groups. We prove that $$ \mathfrak{F} $$ is a π-maximal Fitting class exactly when there is a prime p ∊ π such that the index of the $$ \mathfrak{F} $$ -radical $$ G_\mathfrak{F} $$ in G is equal to 1 or p for every π-subgroup of G. Hence, there exist maximal subclasses in a local Fitting class. This gives a negative answer to Skiba’s conjecture that there are no maximal Fitting subclasses in a local Fitting class (see [1, Question 13.50]).
- Subjects
MAXIMAL subgroups; PRIME numbers; GROUP theory; SOLVABLE groups; MATHEMATICS
- Publication
Siberian Mathematical Journal, 2008, Vol 49, Issue 6, p1124
- ISSN
0037-4466
- Publication type
Article
- DOI
10.1007/s11202-008-0108-7