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%W norm.tex FORMAT documentation B. Eick and C.R.B. Wright
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\Chapter{FNormalizers}
%\index{FNormalizers}
Let $\F$ be an integrated locally defined formation, and let $G$ be
a finite solvable group with Sylow complement basis
%display{tex}
$\Sigma :=
\{ S^p \mid p$ divides $ \|G\| \}$.
%display{html}
%$\Sigma$.
%enddisplay
Let $\pi$ be the set of prime
divisors of the order of $G$ that are in the support of $\F$ and
${\nu}$ the remaining prime divisors of the order of $G$.
Then the *$\F$-normalizer* of $G$ with respect to $\Sigma$ is defined
to be
%display{tex}
$\bigcap_{q \in \nu} S^q \cap
\bigcap_{p \in \pi} N_G( G^{\F(p)} \cap S^p )$.
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%[see the PDF manual].
%enddisplay
The special case $\F(p) = \{ 1 \}$ for all $p$ defines the formation
of nilpotent groups, whose $\F$-normalizers
are the *system normalizers* of $G$. The $\F$-normalizers of a group
$G$ for a given $\F$ are all conjugate. They cover $\F$-central chief
factors and avoid $\F$-hypereccentric ones.
\> FNormalizerWrtFormation( <G>, <F> ) O
\> SystemNormalizer( <G> ) A
If <F> is a locally defined integrated formation in {\GAP} and
<G> is a finite solvable group, then the function `FNormalizerWrtFormation'
returns an <F>-normalizer of <G>. The function `SystemNormalizer' yields a
system normalizer of <G>.
The underlying algorithm here requires <G> to have a special pcgs (see section "ref:Polycyclic Groups" in the {\GAP} reference manual), so the algorithm's first step is
to compute such a pcgs for <G> if one is not known. The complement basis
$\Sigma$ associated with this pcgs is then used to compute the
<F>-normalizer of <G> with respect to $\Sigma$. This process means that
in the case of a finite solvable group <G> that does not have a special pcgs,
the first call of `FNormalizerWrtFormation' (or similarly of `FormationCoveringGroup')
will take longer than subsequent calls, since it will include the
computation of a special pcgs.
The `FNormalizerWrtFormation' algorithm next computes an -system for , a
complicated record that includes a pcgs corresponding to a normal series
of <G> whose factors are either <F>-central or <F>-hypereccentric. A subset
of this pcgs then exhibits the <F>-normalizer of <G> determined by
$\Sigma$. The list `ComputedFNormalizerWrtFormations( <G> )' stores the -normalizers
of <G> that have been found for various formations <F>.
The `FNormalizerWrtFormation' function can be used to study the subgroups of a
single group <G>, as illustrated in an example in Section "Other Applications". In that case it is sufficient to have a function
`ScreenOfFormation' that returns a normal subgroup of on each call.
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