Some criteria for supersolubility of finite groups
A. Vasil'ev, V. Vasil'ev, T. Vasil'eva
All groups considered in this paper are finite.
Let \(H\) be a subgroup of a group \(G\). The permutizer [1, p. 27]
of \(H\) in \(G\) is the
subgroup \(P_G(H)=\langle x\in G\ |\ \langle x\rangle H=H\langle x\rangle \rangle\).
Definition. Let \(H\) be a subgroup of a group \(G\). We say that
\(1)\) \(H\) is permuteral in \(G\), if \(P_G(H)=G\);
\(2)\) \(H\) is strongly permuteral in \(G\) if \(P_U(H)=U\), whenever \(H\leq U\leq G\).
There exist groups that have permuteral but not strongly permuteral subgroups.
For example,
in the group \(G=PSL(2, 7)\), a Sylow
\(3\)-subgroup \(Z_3\) is permuteral in \(G\). Since \(Z_3\leq U\leq G\), where
\(U\) is isomorphic to the alternating group \(A_4\) of degree 4 and \(P_U(Z_3)=Z_3\), it follows that \(Z_3\)
is not strongly permuteral in \(G\).
Theorem 1.Let \(G\) be a metanilpotent group. Then the following statements
are equivalent:
\(1)\) \(G\) is supersoluble;
\(2)\) Every Sylow subgroup of \(G\) is strongly permuteral in \(G\);
\(3)\) Every Sylow subgroup of \(G\) is permuteral in \(G\).
Theorem 2.Let \(G\) be a group. Then the following statements
are equivalent:
\(1)\) \(G\) is supersoluble\(;\)
\(2)\) Every pronormal subgroup of \(G\) is strongly permuteral in \(G\);
\(3)\) Every pronormal subgroup of \(G\) is permuteral in \(G\);
\(4)\) Every Hall subgroup of \(G\) is strongly permuteral in \(G\);
\(5)\) Every Hall subgroup of \(G\) is permuteral in \(G\).
Theorem 3.Let \(G\) be a group. Then the following statements
are equivalent:
\(1)\) \(G\) is supersoluble;
\(2)\) \(G=AB\) is the product of strongly permuteral nilpotent subgroups \(A\) and \(B\) of \(G\);
\(3)\) \(G=AB\) is the product of permuteral nilpotent subgroups \(A\) and
\(B\) of \(G\).
Corollary 3.1.Let \(G\) be a group, and let \(G=AB\) be the product of its Sylow subgroups
\(A\) and \(B\). Then \(G\) is supersoluble if and only if
\(A\) and \(B\) are permuteral in \(G\).
Corollary 3.2.Let \(G\) be a group. Then \(G\) is supersoluble if and only if
\(G=F(G)H\), where \(H\) is a permuteral Carter subgroup of \(G\).
For details, see [2].
References
M. Weinstein (ed.), Between nilpotent and solvable, Passaic: Polygonal Publ. House (1982), 240 p.
A. F. Vasil'ev, V. A. Vasil'ev, T. I. Vasil'eva, On permutizers of subgroups of finite groups,
arXiv:1305.2630v1 [math.GR], May 12 (2013).
