On normal complements of $\mathfrak {F}$-covering subgroups
Author:
H. J. Schmidt
Journal:
Proc. Amer. Math. Soc. 25 (1970), 457-459
MSC:
Primary 20.40
DOI:
https://doi.org/10.1090/S0002-9939-1970-0258960-9
MathSciNet review:
0258960
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Abstract | References | Similar Articles | Additional Information
Abstract: If $\mathcal {F}$ is a suitably restricted formation, we show that an $\mathcal {F}$-covering subgroup $H$ which is a Hall subgroup of the finite, solvable group $G$ is complemented by the $\mathcal {F}$-residual of $G$, provided $H$ normalizes an $\mathcal {F}$-normalizer of $G$. In particular, $H$ is complemented by the $\mathcal {F}$-residual, if $H$ is an $\mathcal {F}$-normalizer of $G$. Further, if $\mathcal {F}$ is the class of nilpotent groups, then $H$ complements the nilpotent residual, if $G$ has pronormal system normalizers. Examples are given to show the necessity of the various hypotheses.
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Additional Information
Keywords:
Formation,
solvable group,
<!β MATH $\mathcal {F}$ β> <IMG WIDTH="22" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$\mathcal {F}$">-covering subgroup,
<!β MATH $\mathcal {F}$ β> <IMG WIDTH="22" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img4.gif" ALT="$\mathcal {F}$">-normalizer,
<!β MATH $\mathcal {F}$ β> <IMG WIDTH="22" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\mathcal {F}$">-residual,
Carter subgroup,
Hall subgroup,
pronormal subgroup,
system normalizer
Article copyright:
© Copyright 1970
American Mathematical Society