Special primitive pairs in finite groups. (Q636743)

This paper is concerned with a special kind of primitive pairs in finite groups. The results in the paper are aimed to play a key role in the author's work towards a new proof of \textit{G. Glauberman}'s \(Z^*\)-Theorem [J. Algebra 4, 403-420 (1966; Zbl 0145.02802)]. Let \(H_1,H_2\) be distinct proper subgroups of \(G\) and \(A\leq H_1\cap H_2\). Let \(\pi\) be the set of prime divisors of \(|A|\) and let \(q\in\pi'\) be a prime. The pair \((H_1,H_2)\) is said to be an \(A\)-special primitive pair of characteristic \(q\) of \(G\) if the following hold: for \(i=1,2\), \(N_G(X)=H_i\), whenever \(1\neq X\) is a normal subgroup of \(H_i\), and \(F^*(H_i)=O_q(H_i)\) (\(F^*(H_i)\) denotes the generalized Fitting subgroup of \(H_i\)); \(C_G(A)\leq H_1\cap H_2\) and \(A\leq Z_\pi^*(H_1)\cap Z_\pi^*(H_2)\) (\(Z_\pi^*(H_i)\) denotes the full pre-image of \(Z(H_i/O_{\pi'}(H_i))\) in \(H_i\)). The main result is the following: Theorem I. Assume that \(A\), \(G\), \(\pi\) and \(q\) are as above. Suppose that \(O_q(G)=1\) and that, whenever \(AC_G(A)\leq H\leq G\), then \(\overline H:=H/O_{\pi'}(H)\) has a unique maximal \(\overline{AC_G(A)}\)-invariant \(q\)-subgroup. If \((H_1,H_2)\) is an \(A\)-special pair of characteristic \(q\) of \(G\) as above and if \(2\in\pi\) or \(q\geq 5\), then \(O_q(H_1)\cap H_2=1=O_q(H_2)\cap H_1\).