%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %W norm.tex FORMAT documentation B. Eick and C.R.B. Wright %% %% 10-30-00 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Chapter{FNormalizers} %\index{FNormalizers} Let $\F$ be an integrated locally defined formation, and let $G$ be a finite solvable group with Sylow complement basis $\Sigma := \{ S^p \mid p$ divides $ \|G\| \}$. Let $\pi$ be the set of prime divisors of the order of $G$ that are in the support of $\F$ and $\overline{\pi}$ the remaining prime divisors of the order of $G$. Then the *$\F$-normalizer* of $G$ with respect to $\Sigma$ is defined to be $\bigcap_{p \in \overline{\pi}} S^p \cap \bigcap_{p \in \pi} N_G( G^{\F(p)} \cap S^p )$. The special case $\F(p) = \{ 1 \}$ for all $p$ defines the formation of nilpotent groups, whose $\F$-normalizers $ \bigcap_{p} N_G( S^p )$ 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 SpecialPcgs), 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 <F>-system for <G>, 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 <F>-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 <G> on each call.