On inner \(\pi\) '-closed groups and normal \(\pi\)-complements (Q1262392)

Let G be a finite group and let \(\pi\) be a set of primes. G is said to be \(\pi\)-homogeneous if \(N_ G(K)/C_ G(K)\) is a \(\pi\)-group for every \(\pi\)-subgroup K of G and weakly \(\pi\)-homogeneous if \(N_ G(P)/C_ G(P)\) is a \(\pi\)-group for every p-subgroup P of G with \(p\in \pi\). If \({\mathfrak P}\) is a class of finite groups, a group G is an inner \({\mathfrak P}\)-group if every proper subgroup of G is a \({\mathfrak P}\)-group but G itself is not a \({\mathfrak P}\)-group. The following results are obtained: 1. If every proper subgroup of the group G is weakly \(\pi\)-homogeneous and weakly \(\pi\) '-homogeneous, then G is either an inner nilpotent group or a direct product of a Hall \(\pi\)- subgroup by a Hall \(\pi\) '-subgroup. 2. If G is weakly \(\pi\)-homogeneous and if one of the following holds, then G has a normal \(\pi\)-complement: i) Every \(\pi\)-subgroup of G is 2-closed. ii) Every \(\pi\)-subgroup of G is 2'-closed. As a consequence of the second result, the following extension of a well- known theorem of Glauberman is obtained: 3. Let G be a finite group and let \(\pi\) be a set of odd primes. Suppose that there exists a prime \(p\in \pi\) and a Sylow p-subgroup P of G such that \(N_ G(Z(J(P)))\) has a normal \(\pi\)-complement. Then G has also one. Also of interest is the following result: 4. G is weakly \(\pi\)-homogeneous iff G is \(\pi\)- homogeneous. Th. 2 extends earlier results of \textit{Z. Arad} and \textit{D. Chillag} [J. Algebra 87, 472-482 (1984; Zbl 0546.20017)] and of \textit{Z. Arad} [Pac. J. Math. 51, 1-9 (1974; Zbl 0287.20014)]. Th. 1 gives an answer to a problem which generalizes problem 2.11 of The Kourovka Notebook [Transl., II. Ser., Am. Math. Soc. 121 (1983; Zbl 0512.20001)].