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 pi-radical of a finite group (here pi 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 < 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.