Towards the Sharp Baer–suzuki Theorem for the π-radical: Symplectic Groups

We study the following conjecture, which is a sharp analogue of the well-known Baer–Suzuki theorem for the π-radical of a finite group. For an arbitrary set π of primes not containing all primes, let r be the smallest prime not in π. Set m = r if r ⩽ 3 and m = r − 1 if r > 3. Then, in a finite group G, the largest normal π-subgroup always coincides with the set of elements x such that any m conjugates of x generate a π-subgroup. To date, this conjecture has been confirmed for any finite group whose every nonabelian composition factor is isomorphic to a sporadic, alternating, linear, or unitary simple group, or to one of the groups ²B₂(q), ²G₂(q), ²F₄(q)′, G₂(q), or ³D₄(q). It is proved that the simple symplectic groups S_2n(q) can be added to this list.