Abstract. We prove that there exists a constant k with the property: if C is a conjugacy class of a finite group G such that every k elements of C generate a solvable subgroup, then C generates a solvable subgroup. In particular, using the Classification of Finite Simple Groups, we show that we can take k = 4. We also present proofs that do not use the Classification Theorem. The most direct proof gives a value of k = 10. By lengthening one of our arguments slightly, we obtain a value of k = 7.

Let G be a finite soluble group that is generated by a conjugacy class consisting of elements of order 3. We show that there exist four conjugates of an element of order 3 that generate a subgroup with the same Fitting height as G. We use this result to find a soluble analogue of the Baer–Suzuki theorem in the case prime 3.

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A group G is called an ATI-group if for any abelian subgroup A of G, A ∩ A x = 1 or A for all x ∈ G. In this paper the finite ATI-groups are classified.

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