6533b7d4fe1ef96bd1262948

RESEARCH PRODUCT

Generation of Certain Matrix Groups by Three Involutions, Two of Which Commute

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Ž . We say that a group is 2, 2 = 2 -generated if it can be generated by three involutions, two of which commute. The problem of determining Ž . which finite simple groups are 2, 2 = 2 -generated was posed by Mazurov w x in 1980 in the Kourovka notebook 3 . An answer to this problem, for some classes of finite simple groups, was given by Ya. N. Nuzhin, namely for w x Chevalley groups of rank 1 in 4 , for Chevalley groups over a field of w x characteristic 2 in 5 , and for the alternating groups and Chevalley groups w x of type A in 6 . In this paper we consider the problem in the more n general context of matrix groups over arbitrary, finitely generated, commutative rings. As a special case of our results, stated in Theorems A, B, and Ž . C below, we obtain that most finite classical groups are 2, 2 = 2 generated, for sufficiently large rank. These groups include Chevalley groups of type B , C , and D in odd characteristic. Our results are n n n constructive, in the sense that we actually define three generating involutions, two of which commute. The choice of groups to include, and the