The establishment.

The following problem was posed by V. D. Mazurov in the Kourovka notebook: 14.69 For every finite simple group, find the minimum number of generating involutions satisfying an additional condition, in each of the following cases: (a) The product of the generating involutions is 1. (b) All generating involutions are conjugate. (c) The conditions (a) and (b) are simultaneously satisfied. (d) All generating involutions are conjugate and two of them commute. This thesis is focused on part (c) of the above problem. For a non-abelian finite simple group, the minimum number of generating involutions with this property must be at least five. Hence, for G, a non-abelian simple group, this thesis approaches the above problem by asking whether G has the following property: 1. G can be generated by five conjugate involutions whose product is 1. Often this is done by asking whether the group G has the following stronger property: 2. G can be generated by three conjugate involutions, with the product of two of them also an involution and conjugate to the other three. After an introductory chapter, this thesis answers these questions for the following simple groups: • The simple alternating groups (chapter 2). Standard results about the structure of the alternating groups are used;