Toward a Sharp Baer–Suzuki Theorem for the π-Radical: Unipotent Elements of Groups of Lie Type

We will look into the following conjecture, which, if valid, would allow us to formulate an unimprovable analog of the Baer–Suzuki theorem for the π-radical of a finite group (here π is an arbitrary set of primes). For an odd prime number r, put m = r, if r = 3, and m = r - 1 if r ≥ 5. Let L be a simple non-Abelian group whose order has a prime divisor s such that s = r if r divides |L| and s > r otherwise. Suppose also that x is an automorphism of prime order of L. Then some m conjugates of x in the group \(\langle L,x\rangle \) $⟨L,x⟩$ generate a subgroup of order divisible by s. The conjecture is confirmed for the case where L is a group of Lie type and x is an automorphism induced by a unipotent element.