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Register a new userThe $3$-closure of a solvable permutation group is solvable
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Let $m$ be a positive integer and let $Omega$ be a finite set. The $m$-closure of $Gleqoperatorname{Sym}(Omega)$ is the largest permutation group on $Omega$ having the same orbits as $G$ in its induced action on the Cartesian product $Omega^m$. The $1$-closure and $2$-closure of a solvable permutation group need not be solvable. We prove that the $m$-closure of a solvable permutation group is always solvable for $mgeq3$.
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Let $V$ be a finite vector space over a finite field of order $q$ and of characteristic $p$. Let $Gleq GL(V)$ be a $p$-solvable completely reducible linear group. Then there exists a base for $G$ on $V$ of size at most $2$ unless $q leq 4$ in which case there exists a base of size at most $3$. The first statement extends a recent result of Halasi and Podoski and the second statement generalizes a theorem of Seress. An extension of a theorem of Palfy and Wolf is also given.
The $2$-closure $overline{G}$ of a permutation group $G$ on $Omega$ is defined to be the largest permutation group on $Omega$, having the same orbits on $OmegatimesOmega$ as $G$. It is proved that if $G$ is supersolvable, then $overline{G}$ can be found in polynomial time in $|Omega|$. As a byproduct of our technique, it is shown that the composition factors of $overline{G}$ are cyclic or alternating of prime degree.
A graph is split if there is a partition of its vertex set into a clique and an independent set. The present paper is devoted to the splitness of some graphs related to finite simple groups, namely, prime graphs and solvable graphs, and their compact forms. It is proved that the compact form of the prime graph of any finite simple group is split.
We obtain non-vanishing of group $L^p$-cohomology of Lie groups for $p$ large and when the degree is equal to the rank of the group. This applies both to semisimple and to some suitable solvable groups. In particular, it confirms that Gromovs question on vanishing below the rank is formulated optimally. To achieve this, some complementary vanishings are combined with the use of spectral sequences. To deduce the semisimple case from the solvable one, we also need comparison results between various theories for $L^p$-cohomology, allowing the use of quasi-isometry invariance.
It is shown that a closed solvable subgroup of a connected Lie group is compactly generated. In particular, every discrete solvable subgroup of a connected Lie group is finitely generated. Generalizations to locally compact groups are discussed as far as they carry.
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