Classical groups, probabilistic methods, and the \((2,3)\)-generation problem.

*(English)*Zbl 0865.20020A group is said to be \((m,n)\)-generated if it is generated by a pair of elements \(x\), \(y\), say, where \(x\) (resp. \(y\)) has order \(m\) (resp. \(n\)). It is known (assuming the classification of finite simple groups) that every such group is generated by a pair of elements. The natural question then arises: to what extent can the orders of such a pair of generators be restricted? This has led to the following: Conjecture. With some exceptions of low rank in characteristic 2 and 3, every finite (non-abelian) simple group is \((2,3)\)-generated.

This represents the “best possible” situation since every \((2,2)\)-generated group is metabelian. The conjecture implies that almost all finite simple groups are images of the modular group \(\text{PSL}_2(\mathbb{Z})\). Using deterministic methods (which (more or less) explicitly provide generators), a number of authors have proved that many finite simple groups are \((2,3)\)-generated. This paper is to date the most important contribution in support of the above conjecture.

The authors use a probabilistic approach which is originally due to Dixon (1969). Let \(G\) be a finite simple classical (or alternating) group. Let \(X\) be the set of all ordered pairs \((x,y)\) in \(G\times G\) where \(x\) (resp. \(y\)) has order 2 (resp. 3) and let \(Y\) be the subset of \(X\) consisting of all such pairs which generate \(G\). Then \(P_{2,3}(G)=|Y|/|X|\) measures the probability that a randomly chosen pair of elements of orders 2 and 3 generate \(G\). The complementary probability \(Q_{2,3}(G)=1-P_{2,3}(G)\) involves those pairs in \(X\) which lie in some maximal subgroup of \(G\).

Using the Classification together with many important results on maximal subgroups, the authors prove that \(Q_{2,3}(G)\) is bounded above (via an absolute constant) by the values of a “zeta function” \(\zeta_G(s)\), for some fixed \(s\). The function \(\zeta_G(s)\), which has only finitely many terms, encodes the indices of the maximal subgroups of \(G\). The authors prove that, for any fixed \(s>1\), \(\zeta_G(s)\to 0\), as \(|G|\to\infty\). The principal results of this paper follow:

Theorem. When \(G\neq\text{PSp}_4(p^k)\), \(P_{2,3}(G)\to 1\), as \(|G|\to\infty\). When \(G=\text{PSp}_4(p^k)\), where \(p>3\), \(P_{2,3}(G)\to 1/2\), as \(|G|\to\infty\). It follows that: Corollary. All but finitely many finite simple classical groups, apart from \(\text{PSp}_4(p^k)\), where \(p=2\) or 3, are \((2,3)\)-generated. The authors also prove that \(\text{PSp}_4(p^k)\) where \(p=2\) or 3, is not \((2,3)\)-generated. To date the only other known infinite family of non-\((2,3)\)-generated finite simple groups is that consisting of the Suzuki groups.

The authors prove similar results using the same method. For example, they show that the probability that a finite simple classical (or alternating) group is generated by three randomly chosen conjugate involutions tends to 1, as \(|G|\to\infty\). It follows then that all but finitely many finite simple classical groups are generated by three conjugate involutions.

Finally the authors combine their probabilistic results with theorems from extremal graph theory to prove, for example, that when \(G\neq\text{PSp}_4(p^k)\) is a finite simple classical (or alternating) group, then there exist subsets \(S\), \(T\) of \(G\), consisting of conjugate involutions and conjugate elements of order 3, respectively, with the property that \(G=\langle s,t\rangle\), for all \(s\in S\), \(t\in T\).

This represents the “best possible” situation since every \((2,2)\)-generated group is metabelian. The conjecture implies that almost all finite simple groups are images of the modular group \(\text{PSL}_2(\mathbb{Z})\). Using deterministic methods (which (more or less) explicitly provide generators), a number of authors have proved that many finite simple groups are \((2,3)\)-generated. This paper is to date the most important contribution in support of the above conjecture.

The authors use a probabilistic approach which is originally due to Dixon (1969). Let \(G\) be a finite simple classical (or alternating) group. Let \(X\) be the set of all ordered pairs \((x,y)\) in \(G\times G\) where \(x\) (resp. \(y\)) has order 2 (resp. 3) and let \(Y\) be the subset of \(X\) consisting of all such pairs which generate \(G\). Then \(P_{2,3}(G)=|Y|/|X|\) measures the probability that a randomly chosen pair of elements of orders 2 and 3 generate \(G\). The complementary probability \(Q_{2,3}(G)=1-P_{2,3}(G)\) involves those pairs in \(X\) which lie in some maximal subgroup of \(G\).

Using the Classification together with many important results on maximal subgroups, the authors prove that \(Q_{2,3}(G)\) is bounded above (via an absolute constant) by the values of a “zeta function” \(\zeta_G(s)\), for some fixed \(s\). The function \(\zeta_G(s)\), which has only finitely many terms, encodes the indices of the maximal subgroups of \(G\). The authors prove that, for any fixed \(s>1\), \(\zeta_G(s)\to 0\), as \(|G|\to\infty\). The principal results of this paper follow:

Theorem. When \(G\neq\text{PSp}_4(p^k)\), \(P_{2,3}(G)\to 1\), as \(|G|\to\infty\). When \(G=\text{PSp}_4(p^k)\), where \(p>3\), \(P_{2,3}(G)\to 1/2\), as \(|G|\to\infty\). It follows that: Corollary. All but finitely many finite simple classical groups, apart from \(\text{PSp}_4(p^k)\), where \(p=2\) or 3, are \((2,3)\)-generated. The authors also prove that \(\text{PSp}_4(p^k)\) where \(p=2\) or 3, is not \((2,3)\)-generated. To date the only other known infinite family of non-\((2,3)\)-generated finite simple groups is that consisting of the Suzuki groups.

The authors prove similar results using the same method. For example, they show that the probability that a finite simple classical (or alternating) group is generated by three randomly chosen conjugate involutions tends to 1, as \(|G|\to\infty\). It follows then that all but finitely many finite simple classical groups are generated by three conjugate involutions.

Finally the authors combine their probabilistic results with theorems from extremal graph theory to prove, for example, that when \(G\neq\text{PSp}_4(p^k)\) is a finite simple classical (or alternating) group, then there exist subsets \(S\), \(T\) of \(G\), consisting of conjugate involutions and conjugate elements of order 3, respectively, with the property that \(G=\langle s,t\rangle\), for all \(s\in S\), \(t\in T\).

Reviewer: A.W.Mason (Glasgow)

##### MSC:

20F05 | Generators, relations, and presentations of groups |

20P05 | Probabilistic methods in group theory |

20D06 | Simple groups: alternating groups and groups of Lie type |

20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |

20E32 | Simple groups |

20E28 | Maximal subgroups |

20D05 | Finite simple groups and their classification |