It is well known that the modular group PSL2(Z) is the free product of a group of order 2 and a group of order 3. Thus the homomorphic images of PSL2(Z) are the groups of order at most 3 and the groups which can be generated by an element of order 2 and an element of order 3. Groups of the latter type are said to be (2,3)-generated. It seems that many (but not all) finite non-abelian simple groups are (2,3)-generated. Thus, for example, G. A. Miller showed in 1901 that the alternating group An is (2,3)-generated for n≥9 and one of the present authors (Tamburini) has shown that SL(n,q) is (2,3)-generated for n≥25 and any prime power q. The results of this paper lengthen the list of finite non-abelian simple groups that are known to be (2,3)-generated. However, the techniques used to investigate the finite groups also extend to linear groups over certain finitely generated commutative rings and, as a consequence, impressive numbers of noteworthy groups, both finite and infinite, are shown to be (2,3)-generated. The following result is proved. Let R be a finitely generated commutative ring, generated by d elements t1,⋯,td, where t1 is a unit of finite multiplicative order. (It is always possible to take t1=1, possibly at the cost of increasing the size of a minimal generating set by 1.) Let En(R) denote the subgroup of SLn(R) generated by the usual elementary matrices. Then En(R) is (2,3)-generated if n≥12d+16. In certain cases, En(R) coincides with SLn(R), for example, if R is either a Euclidean domain, a semilocal ring, or a Hasse domain in a global field, and thus the authors obtain a criterion for SLn(R) to be (2,3)-generated in these three cases. A striking consequence of their result is the fact that SLn(Z) is an epimorphic image of PSL2(Z) for n≥28. A second result of a similar kind concerns certain analogues of the classical symplectic, orthogonal and unitary groups. Suppose that the ring R is as above, generated by d elements, and that n≥12d+25. Suppose in addition that R is one of the three types of ring mentioned in the paragraph above. Then the symplectic group Sp2n(R) is (2,3)-generated, provided that 2 is a unit in R, and the orthogonal group Ω+2n(R) is also (2,3)-generated. If R has a non-trivial involutory automorphism J, together with some hypotheses on the action of J, it is also shown that the special unitary group SU2n(R) is (2,3)-generated. The proofs given in the paper construct explicit generators of order 2 and 3 for the stated groups. The techniques used are to some extent refinements of those used in earlier papers of Tamburini and of Tamburini and Wilson.

### On the (2-3)-generation of some classical groups

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*GAVIOLI, NORBERTO*

##### 1994

#### Abstract

It is well known that the modular group PSL2(Z) is the free product of a group of order 2 and a group of order 3. Thus the homomorphic images of PSL2(Z) are the groups of order at most 3 and the groups which can be generated by an element of order 2 and an element of order 3. Groups of the latter type are said to be (2,3)-generated. It seems that many (but not all) finite non-abelian simple groups are (2,3)-generated. Thus, for example, G. A. Miller showed in 1901 that the alternating group An is (2,3)-generated for n≥9 and one of the present authors (Tamburini) has shown that SL(n,q) is (2,3)-generated for n≥25 and any prime power q. The results of this paper lengthen the list of finite non-abelian simple groups that are known to be (2,3)-generated. However, the techniques used to investigate the finite groups also extend to linear groups over certain finitely generated commutative rings and, as a consequence, impressive numbers of noteworthy groups, both finite and infinite, are shown to be (2,3)-generated. The following result is proved. Let R be a finitely generated commutative ring, generated by d elements t1,⋯,td, where t1 is a unit of finite multiplicative order. (It is always possible to take t1=1, possibly at the cost of increasing the size of a minimal generating set by 1.) Let En(R) denote the subgroup of SLn(R) generated by the usual elementary matrices. Then En(R) is (2,3)-generated if n≥12d+16. In certain cases, En(R) coincides with SLn(R), for example, if R is either a Euclidean domain, a semilocal ring, or a Hasse domain in a global field, and thus the authors obtain a criterion for SLn(R) to be (2,3)-generated in these three cases. A striking consequence of their result is the fact that SLn(Z) is an epimorphic image of PSL2(Z) for n≥28. A second result of a similar kind concerns certain analogues of the classical symplectic, orthogonal and unitary groups. Suppose that the ring R is as above, generated by d elements, and that n≥12d+25. Suppose in addition that R is one of the three types of ring mentioned in the paragraph above. Then the symplectic group Sp2n(R) is (2,3)-generated, provided that 2 is a unit in R, and the orthogonal group Ω+2n(R) is also (2,3)-generated. If R has a non-trivial involutory automorphism J, together with some hypotheses on the action of J, it is also shown that the special unitary group SU2n(R) is (2,3)-generated. The proofs given in the paper construct explicit generators of order 2 and 3 for the stated groups. The techniques used are to some extent refinements of those used in earlier papers of Tamburini and of Tamburini and Wilson.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.