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

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11697/7048
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