Generation of finite almost simple groups by conjugates.(English)Zbl 1037.20016
Let \(G\) be a finite almost simple group and \(L=F^*(G)\). For \(x\in G\), let \(\alpha(x)\) be the minimal number of \(L\)-conjugates of \(x\) which generate the group \(\langle L,x\rangle\). The authors obtain some upper bounds on \(\alpha(x)\). For example, if \(L\) is a simple classical group of dimension at least 5 and \(x\in\operatorname{Aut}(L)\) then \(\alpha(x)\leq n\), unless \(L=\text{Sp}_n(q)\) with \(q\) even, \(x\) is a transvection and \(\alpha(x)=n+1\) (Theorem 4.2).
Reviewer: Victor Mazurov (Novosibirsk)
MSC:
| 20D06 | Simple groups: alternating groups and groups of Lie type |
| 20F05 | Generators, relations, and presentations of groups |
Keywords:
classical groups;exceptional groups;groups of Lie type;finite almost simple groups;Fitting subgroup;numbers of generatorsSoftware:
ATLAS Group RepresentationsCite
Full Text:DOI
References:
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