Orders of intersections of Sylow \(p\)-subgroups in finite groups (Q2709292)

In [``Kourovka Notebook'', 12-th ed., Novosibirsk (1992; Zbl 0831.20003)], V. V. Kabanov posed the following question 5.12. Let \(G\) be a finite group with trivial solvable radical in which there exist two Sylow 2-subgroups with nontrivial intersection. Is it true that if, for any two Sylow 2-subgroups \(P\) and \(Q\) of \(G\) satisfying \(P\cap Q\not=1\), the index \(|P:P\cap Q|\) is at most \(2^n\), then \(|P|\leq 2^{2n}\)? Kabanov gave a solution to this problem for \(n\leq 3\). In the present paper, a more general problem is considered: (**)~Assume that \(G\) is a finite group, \(p\) is a prime, \(O_p(G)=1\), and \(G\) contains two Sylow 2-subgroups with nontrivial intersection. Is it true that if, for any Sylow \(p\)-subgroups \(P\) and \(Q\) of \(G\) satisfying \(P\cap Q\not=1\), the index \(|P:P\cap Q|\) is at most \(p^n\), then \(|P|\leq p^{2n}\)? Using the classification of finite simple groups, the author proves that if the rank of \(P\) is greater than 1, then the answer to (**)~is affirmative. Furthermore, if the rank of \(P\) is 1, then examples are given which indicate that the answer to (**) is negative for all primes \(p\); the indices of all nontrivial intersections in the examples are equal to \(p\), and the orders of these intersections may be arbitrarily large.